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This article is cited in 1 scientific paper (total in 1 paper) Special Bohr–Sommerfeld geometry N. A. Tyurinab a Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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    IntroductionIn his preface to [1] Shafarevich wrote: “Andrey Tyurin chose his mathematical theme very early in his career, already as an undergraduate, and he was to remain faithful to it in essence for his whole life. This chosen theme was the theory of vector bundles and their moduli spaces.” During the last years of his life — so abruptly cut short — Andrey Tyurin published a number of papers devoted to another subject, unusual and even surprising one – at first glance – for an algebraic geometer. All of a sudden, he turned to Lagrangian submanifolds of ordinary real symplectic manifolds and, although not all ideas outlined by him were even brought to reasonable statements, and perhaps only the paper [2] can be mentioned as a complete story, his next goal was crystal clear: to attach the geometry of Largangian submanifolds to algebraic geometry. At that time the deep connection between these two geometries has not manifested itself clearly yet, but [2] was prompted by two essentially different approaches to geometric quantization, the first of which referred directly to algebraic geometry, while the second used the geometry of Lagrangian submanifolds (see the survey part of [2] and the literature cited there). So the key principle – quantization is a universal procedure, and therefore its result must also be universal and independent of the approach taken — dictated the need to consider just this problem. Furthermore, the many years of experience in mathematics suggested that, to solve a problem, you can reduce it to an algebro-geometric one, and then you promptly find a solution. To that time this recipe worked well in many situations: among the most important examples were the calculation of the Donaldson polynomials in terms of the geometry of stable holomorphic vector bundles on algebraic surfaces, then the introduction and development of the Gromov–Witten theory of invariants, and after that a number of constructions of algebraic geometers which implemented the most fantastic intuitions born in string theory. Assumptions made on certain dualities and symmetries were supported by actual constructions, and although universal answers have not been obtained (the conjecture that a Calabi–Yau 3-fold can be foliated by special Lagrangian tori, which could be an embodiment of the motto: “Mirror symmetry is T-duality”, remains a conjecture so far), there is now understanding that mirror symmetry is based on a certain correspondence between the complex and symplectic geometry of Kähler manifolds: by its definition a Kähler manifold has two natures, the complex one and the symplectic one. We can look for such a correspondence at various levels (for instance, in Kontsevich’s homological mirror symmetry we must compare categorical data), however, if you are attracted to more earthly geometric objects, then the following scheme of possible correspondences is implied:
$$
\begin{equation*}
\begin{array}{ccc} \text{complex geometry} & \vert & \text{symplectic geometry} \\ \Downarrow & \vert & \Downarrow \\ \text{complex submanifolds} & \vert & \text{symplectic submanifolds} \\ \text{holomorphic fibrations} & \vert & \text{Lagrangian submanifolds} \\ \end{array}
\end{equation*}
\notag
$$
And while symplectic submanifolds behave in many respects as complex ones (and, in fact, it is natural, in the presence of a complex structure, to consider complex submanifolds as a special — and suitable – class of symplectic submanifolds), the possible link from holomorphic fibrations to Lagrangian submanifolds looks too vague (apart from the simplest case of an elliptic curve). These geometric objects are crucially different in their spaces of possible deformations: a finite-dimensional one in the complex space and a continuum-dimensional one in the second case. However, this is how the nature of a geometric object manifests itself: there is no much sense in the comparison of individual objects because points in a deformation space are very similar as such; the actual nature becomes evident in local deformations. Recall that this was, in fact, the starting point for mirror symmetry, when the deformation space of a complex structure was compared with the deformation space of a Kähler form (for instance, see [3]). In their turn, local deformations give rise to the local structure of appropriate moduli spaces of geometric objects, so that comparing such objects and finding analogies or dualities becomes natural just on the level of moduli spaces, — and this was one of the guiding principles of Andrey Tyurin. However, here we meet the main problem: how can we construct something finite-dimensional from Largrangian submanifolds of a fixed Kähler manifold? At the time when the draft version of A. Tyurin’s last paper [4] was written, two ‘moduli’ constructions were known in Lagrangian geometry: the special Lagrangian geometry of Hitchin and McLean (see [5]), with finite-dimensional moduli spaces, and abelian Lagrangian algebraic geometry (ALAG) from [2], with infinite-dimensional ones. The range of applications of the first reduces to the case of Calabi–Yau manifolds, which is perhaps satisfactory from the point of view of string theory. The second geometry is more universal: the authors of [2] put focus on arbitrary simply connected compact algebraic varieties because it was important in their constructions that the Kähler form be integer. The two theories turned out to be mutually transversal: in an algebraic Calabi–Yau manifold the family of special Lagrangian submanifolds can contain only a discrete (and in fact, finite) subset of elements satisfying the Bohr–Sommerfeld condition, which can be used in calculations of certain integer-valued invariants. For all of this, both constructions could naturally be fit into an algebro-geometric context: the first construction by the introduction of the structure of supercyles and the second by the introduction of half-weights, so that moduli spaces in both classes turned to complex spaces. That is, both examples fit into A. Tyurin’s principal strategy of finding a moduli space points in which are certain special Lagrangian submanifolds or classes of such manifolds, which is an algebraic variety or is close to it. Unfortunately, the final version of [4] was not prepared by Andrey Tyurin himself: it was finalized on the basis of sketches of slides of his last talk at the G. Fano memorial conference and some fragments of manuscripts; we can only guess where Tuyrin’s marvelous geometric intuition could take him. However, the general problem he clearly stated — constructing, for a sufficiently wide class of algebraic varieties, finite-dimensional moduli spaces points in which are Lagrangian submanifolds, by means of purely mathematical methods, without referring to string theory of something outside the standard course — is interesting and important. Special Bohr–Sommerfeld geometry, the main definitions and constructions of which can be found in [6]–[10] and other papers, is an attempt to advance in this direction. Many conjectures that arose in the process, turned out to be too crude or simplistic in comparison with the actual geometric picture, connections with important related areas were not immediately visible, and some dead-ends of the construction turned out surprisingly into growing points of further prospects. For this reason, here we do not follow the pattern of the papers mentioned above, nor reproduce many details and particularities, but we look at constructions from new points of view, putting great emphasis on aspects that can perhaps be helpful in identifying further ways for the implementation of the agenda indicated in [4]. 1. General theoryConsider an arbitrary compact simply connected symplectic manifold $(M,\omega)$ of real dimension $2n$ such that the symplectic form $\omega$ defines an integer cohomology class. Then there is a pair $(L,a)$ over $M$ which consists of a complex line bundle $L \to M$ with fixed Hermitian structure and a Hermitian connection $a \in \mathcal{A}_h(L)$ with curvature form $F_a=2\pi \imath \omega$. In geometric quantization the pair $(L,a)$ is called prequantization data; in the simply connected case the connection $a$ is defined uniquely up to a gauge transformation, and it is clear from the definitions that $c_1(L)=[\omega]$. Since we use $a$ as a parameter in what follows, we use the following notation for the corresponding orbit:
$$
\begin{equation}
\{a \in \mathcal{A}_h(L) \colon F_a=2 \pi \imath \omega\}= \mathcal{O}_{\omega} \subset \mathcal{A}_h(L);
\end{equation}
\tag{1}
$$
abusing the language we will sometimes call the selection of a particular $a \in \mathcal{O}_{\omega}$ the choice of a gauge. By construction $\mathcal{O}_{\omega}$ is isomorphic to the quotient of the gauge group $\mathcal{G}_h(L)$ by the structure group $\mathrm{U}(1)$, because the stabilizer of a connection $a$ consists precisely of the constant gauge transformations. We say that a Lagrangian submanifold $S \subset M$ satisfies the Bohr–Sommerfeld condition (or is a BS-Lagrangian submanifold for short) if the restriction $(L,a)\big|_S$ has a covariantly constant section $\sigma_S$, which is unique up to multiplication by a constant. It is easy to see that the BS-property is independent of the choice of a gauge because it can be stated as follows in the invariant form: $S$ is a BS-submanifold if and only if for each loop $\gamma \subset S$ and any disc $D_{\gamma} \subset M$ bounded by it the symplectic area $\displaystyle\int_{D_{\gamma}} \omega$ is an integer (recall that $M$ under consideration is simply connected, so such a disc exists for each $\gamma$). The idea of the additional constraint in the form of the special BS-condition (SBS-condition for short) arose from the following question. Let $S_1,S_2 \subset M$ be two BS-submanifolds intersecting trasversally in a finite set of points: $S_1 \cap S_2=\{p_1,\dots,p_m\}$. Can we impose an additional condition on $S_i$ so that for each pair $p_j,p_k\in S_1 \cap S_2$ of points in the intersection and each pair of arcs $\gamma_i \subset S_i$, $i =1,2$, joining these points along each $S_i$ the symplectic area of any disc $D$ with boundary $\partial D=\gamma_1 \cup \overline \gamma_2$ is zero: $\displaystyle\int_D \omega=0$ (of course, we only consider pairs $\gamma_i$ for which such discs do exist)? The geometric meaning of this question is connected with the definition of Floer cohomology: if the symplectic area is zero, then for no almost complex structure compatible with $\omega$ can there exist holomorphic films with vertices $p_j$ and $p_k$, so the Floer cohomologies for the pair $S_1$, $S_2$ must be trivial. A partial answer to this question can be given in terms of the SBS-property. Definition 1. A BS-submanifold $S \subset M$ is said to be special with respect to a smooth section $\alpha \in \Gamma (M,L)$ of the prequantization bundle (or satisfies the $ \alpha$- SBS-condition for short) if the restriction $\alpha\big|_S$ has the form $e^{\imath c} f \sigma_S$, where $c \in \mathbb{R}$ is a real constant, $f \in C^{\infty}(S,\mathbb{R}_+)$ is a positive smooth function on $S$, and $\sigma_S$ is a covariantly constant section of the restriction $(L,a)\big|_S$. This definition explains the attribute ‘special’ in the name of our subject because, for instance, in the SpLAG programme [5] being ‘special’ means that certain higher-order forms are proportional with a constant coefficient, as are the sections $\alpha\big|_S$ and $\sigma_S$ in Definition 1. However, to solve the above problem of zero symplectic area of films spanning the points of intersection of $S_1$ and $S_2$, we must re-formulate the above definition as follows. For arbitrary smooth sections in $\Gamma(M,L)$ consider the following representation defined for a fixed gauge, that is, in the presence of a connection $a \in \mathcal{O}_{\omega}$:
$$
\begin{equation}
\rho\colon \alpha \mapsto \rho(\alpha)=\frac{1}{2\pi}\, \frac{\langle\nabla_a\alpha,\alpha\rangle_h}{\langle\alpha,\alpha\rangle_h} \in\Omega^1_{M \setminus D_{\alpha}} \otimes \mathbb{C},
\end{equation}
\tag{2}
$$
where $D_{\alpha}=\{\alpha=0\}$ is the set of zeros of the section $\alpha$. By Theorem 2 in [6] Definition 1 above can concisely be formulated in terms of this $\rho$-representation. Definition 1'. A smooth $n$-dimensional submanifold $S \subset M$ satisfies the $\alpha$-SBS-condition with respect to the smooth section $\alpha \in \Gamma(M,L)$ if and only if In fact, from this condition we deduce first of all that $S$ is Lagrangian: if the restriction of $\operatorname{Im} \rho(\alpha)$ to $S$ vanishes, then the differential of this 1-form vanishes too, and this differential is proportional to the symplectic form (for further details of the proof of the equivalence of two defintions, see [6]). Remark 1. Definition 1' suggests a natural generalization of the SBS-condition to Lagrangian immersions and, more generally, singular varieties, which expands the scope of SBS-geometry significantly. In addition, the defining condition $\operatorname{Im}\rho(\alpha)\big|_S \equiv 0$ is of calibration type in the fashion of the general theory in [11]: the 1-form $\operatorname{Im}\rho(\alpha)$ is here the calibration, just as in the special Lagrangian geometry of Hitchin and McLean the calibration form is the imaginary part of the highest-order holmorphic form on the Calabi–Yau manifold (see [5]). Now we return to the problem of two BS-Lagrangian submanifolds $S_1$ and $S_2$ stated above. It is obvious that there exist smooth sections $\alpha_1,\alpha_2 \in \Gamma(M,L)$ such that $S_i$ is an $\alpha_i$-SBS-submanifold. In this case the above condition is satisfied if there exists a section $\alpha$ such that both $S_1$ and $S_2$ are $\alpha$-SBS-submanifolds. In fact, in this case from Definition 1' we obtain
$$
\begin{equation*}
\operatorname{Im}\rho(\alpha)\big|_{S_1}= \operatorname{Im} \rho(\alpha)\big|_{S_2} \equiv 0,
\end{equation*}
\notag
$$
so the integral of the 1-form $\operatorname{Im} \rho(\alpha)$ over the contour $\gamma_1 \cup \overline \gamma_2$ must be zero, and by Stokes’ formula we have for each film sealing the hole betweeb $S_1$ and $S_2$ with corners at $p_j$ and $p_k$. We stress that here we only partially solve the problem about a pair of Lagrangian submanifolds. The above $\rho$-representation relates our constructions to the theory of Weinstein manifolds, a topical subject in symplectic geometry. The main property of the 1-form $\rho(\alpha)$ is as follows (see details in [6]): its real part $\operatorname{Re}\rho(\alpha)$ is exact and its imaginary part satisfies the equation ${\rm d}\operatorname{Im}\rho(\alpha)=\omega$ in the complement $M \setminus D_{\alpha}$. Namely,
$$
\begin{equation}
\operatorname{Re} \rho(\alpha)=\frac{1}{4\pi}\, \frac{\langle\nabla_a \alpha,\alpha\rangle_h+ \langle\alpha,\nabla_a\alpha\rangle_h}{\langle\alpha,\alpha\rangle_h}= \frac{1}{4 \pi}\,\frac{{\rm d}(\langle\alpha,\alpha\rangle_h)} {\langle\alpha,\alpha\rangle_h}\,,
\end{equation}
\tag{3}
$$
which is $(2\pi)^{-1}\,{\rm d}\log|\alpha|_h$; and the imaginary part induces the vector field
$$
\begin{equation}
\lambda(\alpha)=\omega^{-1}\bigl(\operatorname{Im}\rho(\alpha)\bigr),
\end{equation}
\tag{4}
$$
which is a Liouville field, that is,
$$
\begin{equation*}
\mathcal{L}_{\lambda(\alpha)}\omega=\omega\quad\text{on}\ M \setminus D_{\alpha}.
\end{equation*}
\notag
$$
One and the same symplectic manifold $M \setminus D_{\alpha}$ can carry different Liouville fields, and the most intersecting case is when a Liouville field $\lambda$ admits a compatible Lyapunov function $\Phi$. In this case we say that a Weinstein structure $(\lambda, \Phi)$ is defined on an open symplect manifold (see [12]). It is known that if the section $\alpha$ is pseudoholmorphic with respect to some almost complex structure compatible with $\omega$, then the Lyapunov function for the corresponding field $\lambda(\alpha)$ has the simple form In general, it could be interesting to find a condition on a section $\alpha$ which ensures that the corresponding Liouville field $\lambda(\alpha)$ can be complemented to a Weinstein structure on $M \setminus D_{\alpha}$. For further reasoning we need the main construction from [2]; in the situation under consideration it is the infinite-dimensional Frechét-smooth moduli space of Bohr–Sommerfeld Lagrangian cycles of fixed topological type, which is usually denoted by $\mathcal{B}_S$ (the construction and study of such moduli spaces makes up an area of mathematics which can be called Bohr–Sommerfeld geometry, of BS-geometry for short, so the topic of our paper consists in constructing an appropriate arcade from BS to SBS). Points of this moduli space can be represented by BS-Lagrangian submanifolds of fixed topological type $\operatorname{top}S$ and homology class $[S] \in H_n(M,\mathbb{Z})$. Local charts for $\mathcal{B}_S$ are given by Darboux–Weinstein neighbourhoods: each BS-submanifold $S \subset M$ has a tubular neighbourhood $\mathcal{O}_{\rm DW}(S)$ that is symplectomorphic to a small neighbourhood of the zero section in $T^*S$, and all Bohr–Sommerfeld Lagrangian submanfiolds sufficiently close to $S$ and of the same topological type can be represented as the graphs of exact forms in this neighbourhood (for details, see [2]). Since $\rho(c\alpha)=\rho(\alpha)$ for each $c \in \mathbb{C}^*$, the $\rho$-representation depending on the choice of the gauge $a \in \mathcal{O}_{\omega}$ associates complex 1-forms, in place of sections in $\Gamma(M,L)$, with classes of sections in the projectivization $\mathbb{P}(\Gamma(M,L))$, so that below we can also use the notation $\rho(p)$ for $p \in \mathbb{P}(\Gamma(M,L))$. Here is an illustration of yet another property of the $\rho$-representation of classes of smooth sections: given a class $p \in \mathbb{P}(\Gamma(M,L))$ represented by a smooth section $\alpha \in \Gamma(M,L)$ with set of zeros $D_p$ (we stress that this set depends only on $p$, rather than on the choice of $\alpha$), consider the subset $\Delta_p \subset \mathcal{B}_S$ defined by the condition $D_p \cap S \ne \varnothing$. The complement $\mathcal{B}_S \setminus \Delta_p$ is a system of H-connected components $K_i$, where by H-connectivity we mean the capability to connect any data $S_0,S_1 \subset M \setminus D_p$ by a family of Hamiltonian deformations $S_t$ so that $S_t \cap D_p= \varnothing$ for each $t \in [0,1]$. The form $\operatorname{Im}\rho(\alpha)\big|_S$ is closed on $S$ and, remarkably, the cohomology class $[\operatorname{Im}\rho(\alpha)]\in H^1(S, \mathbb{Z})$ is integer for each $S \in K_i$ in each H-connected component $K_i$. In fact (see [6]), this class can be realized as follows: the coefficient in the equality $\alpha\big|_S=\psi(\alpha,S)\sigma_S$ defines a map $\psi\colon S \to \mathbb{C}^*$, and the class in question is just $\psi^*h$, where $h$ is the generator of the first cohomologies in $H^1(\mathbb{C}^*,\mathbb{Z})$. Hence the class $[\operatorname{Im}\rho(\alpha)]$ is constant on each component $K_i$, and for the existence of an SBS-submanifold in $K_i$ it is obviously necessary that this class is trivial on $K_i$. Now let the corresponsing class of cohomologies be trivial on some $K_i$; then the $\rho$-representation $\operatorname{Im}\rho(\alpha)$ is an exact form on each $S \in K_i$. However, by [2] an exact form is a tangent vector in $T_S \mathcal{B}_S$, which gives us a realization of $\operatorname{Im}\rho(\alpha)$ by a smooth vector field on $K_i$ such that its zeros are just SBS-submanifolds with respect to the section $\alpha$. Definitions 1 and 1' presented above enable us to construct in the Cartesian product $\mathbb{P}(\Gamma(M,L)) \times \mathcal{B}_S$ a certain ‘incidence cycle’ related to the SBS-condition and consisting of the pairs $(p,S)$ such that $S$ is $\alpha$-SBS with respect to the section $\alpha$ corresponding to the point $p \in \mathbb{P}(\Gamma(M,L))$. Below we denote this cycle by $\mathcal{U}_{\rm SBS}(a)$. Since this is a subset of a Cartesian product, we have the two canonical projections
$$
\begin{equation}
q_1\colon \mathcal{U}_{\rm SBS}(a) \to \mathbb{P}(\Gamma(M,L)) \quad\text{and}\quad q_2\colon \mathcal{U}_{\rm SBS}(a) \to \mathcal{B}_S,
\end{equation}
\tag{5}
$$
whose properties interest us first of all. We begin by noticing that $q_1$ is invariant under gauge transformations. The incidence cycle $\mathcal{U}_{\rm SBS}(a)$ depends of the connection selected in the orbit, and the action of the gauge group $\mathcal{G}_h(L)$ has the representation
$$
\begin{equation}
g(\mathcal{U}_{\rm SBS}(a))=\mathcal{U}_{\rm SBS}(g(a)).
\end{equation}
\tag{6}
$$
At the same time $\mathcal{G}_h(L)$ acts in the standard way on the space of sections $\Gamma(M,L)$ and acts projectively on $\mathbb{P}(\Gamma(M,L))$. Since the actions of $\mathcal{G}_h(L)$ on the spaces $\mathcal{A}_h(L)$ and $\Gamma(M,L)$ agree, we can immediately see that the canonical projection $q_1$ is equivariant. Thus, all incidence cycles $\mathcal{U}_{\rm SBS}(a)$ are conjugate by the action of the gauge group and have similar properties independently of the particular choice of a gauge. On the other hand, if $S \subset M \setminus D_p$ is a BS-submanifold, then it is natural to ask if there exists a global smooth section with a given set of zeros $D_{\alpha'}=D_p$ and such that $S$ is an $\alpha'$-SBS-submanifold. The set $D_p$ appeared above as the set of zeros of some section $\alpha$ whose class corresponds to $p \in \mathbb{P}(\Gamma(M,L))$. We defined above an integer class of cohomologies $[\operatorname{Im}\rho(\alpha)] \in H^1(S,\mathbb{Z})$ such that when it is trivial, we have a representation
$$
\begin{equation}
\operatorname{Im}\rho(\alpha)\big|_S={\rm d}f, \qquad f \in C^{\infty}(S,\mathbb{R}),
\end{equation}
\tag{7}
$$
where the smooth function $f$ can be extended to a smooth global function $F \in C^{\infty}(M,\mathbb{R})$, while the gauge transformation $g_F=e^{-\imath F}$ takes $\alpha$ to $g_F(\alpha)=\alpha'$ such that this $S$ is an $\alpha'$-SBS-submanifold. We can regard the set of zeros $D_p$ as an invariant of an orbit of the action of $\mathcal{G}_h(L)$ on $\mathbb{P}(\Gamma(M,L))$, and the next question arising naturally in the framework of our discussion is whether or not from the pair of disjoint variaties $(D_p,S)$ we can determine the class $[\operatorname{Im}\rho(\alpha)]$ introduced above. We can give the following definition. Definition 2. A Lagrangian submanifold $S$ of a symplectic manifold $(M,\omega)$ is said to be $D$-exact with respect to a (maybe singular) subvariety $D_p$ representing a homology class $[D_p]=\mathrm{P.D.}[\omega]$ if for each loop $\gamma \subset S$ there exists a disc $B \subset M$, $\partial B=\gamma$, such that The following result can easily be stated in these terms. Proposition 1. A BS-submanifold $S \subset (M,\omega)$ is $\alpha$-SBS with respect to a section $\alpha$ in the class of gauge-equivalent sections with set of zeros $D_p$ if and only if $S$ is $D$-exact with respect to $D_p$. In fact, as we saw above, the required property is equivalent to the exactness of the restriction $\operatorname{Im}\rho(\alpha)\big|_S$. In turn, this is equivalent to the vanishing integral $\displaystyle\int_{\gamma}\operatorname{Im}\rho(\alpha)$ over each loop $\gamma \subset S$. On the other hand the differential of this 1-form is equal to $\omega$, and after using Stokes’ formula on a disc $B \subset M$ bounded by this loop it remains to add expressions for the residues at points in the intersection $B \cap D_p$, whose contribution corresponds to the topological intersection index of these two submanifolds. However, while in the transition to equivalence classes modulo the action of $\mathcal{G}_h(L)$ smooth sections are replaced by their zero subvarieties, the SBS-correspondence suggests a certain equivalence relation between BS-submanifolds. In the light of Proposition 1 this relation must preserve $D$-exactness, so the ‘dual’ gauge group must be the group $\operatorname{Ham}(M \setminus D_p)$ of Hamiltonian isotopies of the complements. In fact, if $S_0$ is a $D$-exact submanifold with respect to some set of zeros $D_p$ and $\phi^t_{X_H}(S_0)=S_t$ is a Hamiltonian isotopy generated by a smooth function $H$ which is a constant in a neighbourhood of $D_p$, the Lagrangian submanifold $S_t$ must be BS and must lie in the complement $M \setminus D_p$. However, $D$-exactness is stable under continuous deformation, so all the $S_t$ must be $D$-exact. Thus, the canonical projection $q_1$ has the following natural factorization:
$$
\begin{equation}
\mathcal{U}_{\rm SBS} \to \mathbb{P}(\Gamma(M,L))\to \{D_p\},
\end{equation}
\tag{8}
$$
where the last map is in its turn the canonical projection of the principal $\mathcal{G}_h(L)$-bundle over the base $\{D_p\}$ consisting of all possible sets of zeros of smooth sections of $L$. Then there is a quotient correspondence
$$
\begin{equation}
\mathcal{F}\colon \mathcal{U}_{\rm SBS} \to \{(D_p,\langle S \rangle)\},
\end{equation}
\tag{9}
$$
assigning to a pair $(p,S) \in \mathcal{U}_{\rm SBS}$ the zero subvariety $D_p$ and the class of $\operatorname{Ham}(M \setminus D_p)$-equivalent $D$-exact Lagrangian submanifolds of fixed topological type. This last formula is universal from the point of view of gauges: choosing a particular gauge $a \in \mathcal{O}_{\omega}$ we select a concrete representative of the space of $\operatorname{Ham}(M \setminus D_p)$-equivalent $D$-exact Lagrangian submanifolds of the complement. Moreover, given a section of the principal $\mathcal{G}_h(L)$-bundle $\mathbb{P}(\Gamma(M,L))\to\{D_p\}$, we obtain a section of the fibration $\mathcal{F}$. The map $\mathcal{F}$ will be important in our constructions below. We turn to the analytic properties of the canonical projection $q_1$. First of all, we notice that the image $q_1(\mathcal{U}_{\rm SBS}(a))$ cannot coincide with the projective space $\mathbb{P}(\Gamma(M,L))$: if we take a section with set of zeros $D_p=M \setminus \mathcal{O}_{\epsilon}(x)$, where $\mathcal{O}_{\epsilon}(x)$ is a small neighbourhood of a point $x \in M$, then the preimage is empty. In fact, the BS-submanifold has integer periods, so it cannot be placed in a small neighbourhood of a point. On the other hand the following result holds. Proposition 2. For any choice of the gauge the canonical projection $q_1$ has discrete fibres; its image $\operatorname{Im}q_1 \subset \mathbb{P}(\Gamma(M,L))$ is open; its differential ${\rm d}q_1$ has a trivial kernel. We briefly recall the proof (for details, see [6]). Let $(p_0,S_0) \in \mathcal{U}_{\rm SBS}(a)$. By definition this means that $\operatorname{Im}\rho(p_0)\big|_{S_0}=0$. Let $S_t$ be a deformation formed by BS-submanifolds with the same property. Then, by the general theory of Darboux–Wienstein neighbourhoods, starting from some small $t_0$, all $S_t$ must have the representation $\phi^t_{X_H}S_0$, where $H$ is a smooth function in a neighbourhood of $S_0$ (which can depend on $t$), $X_H$ is the corresponding Hamiltonian vector field, and $\phi^t_{X_H}$ is the flow generated by this field. However, in this case it is impossible that the restrictions $\operatorname{Im}\rho(p)\big|_{S_t}$ vanish as required. Moreover, as shown in [6], a Darboux–Weinstein neighbourhood of $S_0$ cannot contain another Lagrangian submanifold $S_1$ satisfying the SBS-condition with respect to the same class $p$ of sections of the prequantization bundle. In fact, this also yields the third required assertion: the kernel of the differential of the canonical projection must consist of pairs of infinitesimal deformations $(\delta p_0,\delta S_0)$ such that the first element in it is trivial. However, since infinitesimal deformations $\delta S_0$, as tangent vectors to the moduli space $\mathcal{B}_S$ at the point $S_0$, correspond to Hamiltonian deformations of $S_0$ (see [2]), the same arguments show that $\delta S_0$ must also be trivial. Thus, the canonical projection
$$
\begin{equation*}
q_1\kern-1pt\colon \mathcal{U}_{\rm SBS}(a) \to \mathbb{P}(\Gamma(M,L))
\end{equation*}
\notag
$$
has the structure of a multiple covering over its image $\operatorname{Im}q_1$, which is an open subset of $\mathbb{P}(\Gamma(M, L))$. The projective space $\mathbb{P}\Gamma(M,L)$ carries the induced Kähler structure: the complex space $\Gamma(M,L)$ has the Hermitian structure
$$
\begin{equation}
\langle\alpha_1,\alpha_2\rangle=\int_M(\alpha_1,\alpha_2)_h\,{\rm d}\mu,
\end{equation}
\tag{10}
$$
where ${\rm d}\mu$ is the Liouville form induced by the symplectic form $\omega$. In projectivization, this gives rise to the corresponding Fubini–Study Kähler metric with Kähler form $\Omega_{\rm FS}$. Proposition 2 ensures that by pulling back this Kähler structure the incidence cycle $\mathcal{U}_{\rm SBS}(a)$ acquires a Kähler structure with Kähler form $q_1^* \Omega_{\rm FS}$. This form has no kernel on the infinite-dimensional space, but nevertheless it does not define an isomorphism between the tangent and contangent spaces, that is, it is Kähler in a weak sense. Note that by construction this structure is independent of all additional choices (apart from the fundamental choice of a gauge). Thus, starting from a general symplectic situation, we obtain an object of Kähler geometry. Next we turn to the second canonical projection $q_2\colon \mathcal{U}_{\rm SBS} \to \mathcal{B}_S$. The following result holds: $q_2$ is factorized by means of the map
$$
\begin{equation}
\tau\colon \mathcal{U}_{\rm SBS} \to T \mathcal{B}_S, \qquad \tau(p,S)=\bigl(S,\rho(\alpha)\big|_S\bigr).
\end{equation}
\tag{11}
$$
Recall (see [2]) that tangent vectors at a point $\mathcal{B}_S$ represented by a submanifold $S$ are exact forms ${\rm d}f \in \Omega^1_S$, $f \in C^{\infty}(S,\mathbb{R})$; by the properties of the $\rho$-representation its real part $\operatorname{Re}\rho(\alpha)$ is an exact form independent of the gauge since it is the logarithmic derivative of the pointwise modulus of the section $|\alpha|_h$ (see [6]). It is easy to see that after the introduction of $\tau$ we can factorize the second canonical projection as follows:
$$
\begin{equation}
q_2=\pi \circ \tau, \qquad \pi\colon T \mathcal{B}_S \to \mathcal{B}_S,
\end{equation}
\tag{12}
$$
and the following result is important here. Proposition 3. Fibres of the map $\tau$ are Kähler in $\mathcal{U}_{\rm SBS}$ with respect to the Kähler structure pulled back from $\mathbb{P}(\Gamma(M,L))$. In fact, for an arbitrary pair $(S,{\rm d}f) \in T_S\mathcal{B}_S$ the fibre $\tau^{-1}(S,{\rm d}f) \subset \mathcal{U}_{\rm SBS}$ consists precisely of the pairs $(p,S)$ such that there exists a section $\alpha$ representing the class $p$ such that
$$
\begin{equation*}
\alpha\big|_S=e^{f+z}\sigma_S,\qquad z \in \mathbb{C}.
\end{equation*}
\notag
$$
In the Hilbert space $\Gamma(M,L)$ this distinguishes the linear subspace
$$
\begin{equation}
\Lambda(S,{\rm d}f)=\bigl\{\alpha \in \Gamma(M,L) \colon \alpha\big|_S=C(e^f\sigma_S),\ C \in \mathbb{C}^*\bigr\}
\end{equation}
\tag{13}
$$
punctured at zero, so that
$$
\begin{equation*}
q_1(\tau^{-1}(S,{\rm d} f))=\mathbb{P}(\Lambda(S,{\rm d}f)),
\end{equation*}
\notag
$$
and since the above projective space is obviously Kähler in $\mathbb{P}(\Gamma(M,L))$ and the Kähler structure on $\mathcal{U}_{\rm SBS}$ is pulled back from the base of the projection $q_1$, we obtain the required result.
2. Case of an algebraic manifold: thesisIf to the general framework of our constructions we add some further natural data, namely, a complex structure $I$ on $M$ compatible with the symplectic form $\omega$, then we obtain something more than a mere Kähler manifold, because, as agreed above, the Kähler form $\omega$ defines an integer cohomology class. This means the existence of a holomorphic bundle $L_I \to M_I$ (which is topologically just our prequantization bundle) some tensor power of which gives rise to an embedding of $M_I$ in a projective space (see [13]). On the other hand, if $X$ is a projective algebraic variety, then by definition there exists an ample linear bundle $L \to X$, so that $c_1(L)$ is realized by a symplectic $(1,1)$-form with respect to the complex structure, and we obtain initial data for our constructions (recall that an important assumption in our construction is that $X$ is simply connected). The resulting symplectic form is not uniquely defined for a fixed ample bundle $L$, but it is easy to see that two forms in the same class induce equivalent geometries of Lagrangian submanifolds. On the other hand, for proportional classes geometries can be different. For example, in the simplest case when $X=\mathbb{C}\mathbb{P}^1$ and the two bundles are $L=\mathcal{O}(1)$ and $L^2=\mathcal{O}(2)$, our constructions result in fundamentally different answers, because for $L$ there are no smooth BS-Lagrangian submanifolds, so that the SBS-construction produces a trivial result, while the result for $L^2$ is non-trivial. Furthermore, the following example is appropriate here. Let the algebraic manifold $X=\mathbb{C} \mathbb{P}^1 \times \mathbb{C} \mathbb{P}^1$ be endowed with the polarizations $L_1=\mathcal{O}(1,1)$ and $L_2=\mathcal{O}(1,2)$, which generate the Kähler forms
$$
\begin{equation*}
\omega_1=\omega_{\rm FS} \oplus \omega_{\rm FS}\quad\text{and}\quad \omega_2=\omega_{\rm FS} \oplus 2 \omega_{\rm FS}
\end{equation*}
\notag
$$
(we drop the indications of the canonical projections onto direct factors and some similar signs for short). As regards $\omega_1$, there exists a Lagrangian sphere $S^2$ smoothly embedded by the antidiagonal map $\{[x_0:x_1] \times [\overline x_0:\overline x_1]\} \subset X$, where $[x_0:x_1] \in \mathbb{C} \mathbb{P}^1$, so the homology class $(1,1) \in H_2(X,\mathbb{Z})$ can be realized by a Lagrangian embedding. However, it is obvious that the same class cannot be realized by a Lagrangian submanifold with respect to $\omega_2$; moreover, the existence of a Lagrangian submanifold of this type is not clear for the second form. Thus, rather than a mere complex manifold $X$, we consider here a pair $(X,L)$, where $L$ is an ample bundle. The conjectural resulting finite-dimensional moduli space will depend on the choice of this integer parameter. Remark 2. It is worth mentioning here a similar choice required for the specification of the moduli space of holomorphic bundles of fixed topological type on an algebraic variety (see [1]). The (semi)stability conditions imposed on the bundle depends on the choice of a polarization, that is, on just the same data as in our case. We stress that, so far, this analogy has found no reasonable explanation. Recall another construction, which allows us to construct a symplectic (Kähler) form on an algebraic variety $X$ with a fixed very ample bundle $L \to X$. We select some suitable Hermitian structure $h$ on $L$; then for each holomorphic section $\alpha \in H^0(X,L)$ we can define the function which is smooth in the complement $X \setminus D_{\alpha}$ and defines a Kähler potential there. In other words, the 2-form $\omega(\alpha)={\rm d}I\,{\rm d}\Psi_{\alpha}$ is a non-degenerate closed form of type $(1,1)$ defining the Kähler structure on the complement $X \setminus D_{\alpha}$. Moreover, ampleness ensures that such a Hermitian structure exists and there are sufficiently many holomorphic sections to globalize the local forms $\omega(\alpha_i)$ to a global Kähler form $\omega_h$ (see [13]).On the other hand the choice of a Hermitian structure $h$ and a Kähler form $\omega_h$ specifies a unique Hermitian connection $a_I \in \mathcal{A}_h(L)$ with curvature form proportional to $\omega_h$ and such that the corresponding operator $\overline\partial_{a_I}$ is precisely the $\overline \partial$-operator defining the existing complex structure on $L$. Thus, given a pair $(X,L)$, the choice of a suitable Hermitian structure $h$ on $L$ produces a compatible symplectic form $\omega_h$ and a distinguished gauge $a_I \in \mathcal{O}_{\omega_h}$, and the correspondence $\alpha \mapsto \Psi_{\alpha}$ producing these data is very important for our constructions. Now, by Proposition 2 we have a canonical projection $q_1$ with discrete fibres from (5). Hence if in (5) we have a finite-dimensional projective space, then the source space is supposed to be finite dimensional. In the algebraic case, given a pair $(X,L)$, the natural candidate is $\mathbb{P}(H^0(X,L))$, so we arrive at the following statement. Thesis. Let $(X,L)$ be a simply connected smooth projective variety with an ample bundle on it. Then the moduli space of SBS-Lagrangian submanifolds of fixed topological type can be defined to be the space where the symplectic data are produced by a suitable Hermitian structure $h$ on $L$, the topological type of Lagrangian submanifolds is specified by the data $([S] \in H_n(X,\mathbb{Z}),\operatorname{top}S)$, and the gauge $a_I$ is distinguished by the combination of $h$ and the original complex structure. Thus, for the definition of an appropriate finite-dimensional space in the infinite-dimensional space $\mathcal{U}_{\rm SBS}$ in the algebro-geometric situation, we must fix only a suitable Hermitian structure! This determines simultaneously the form, prequantization data, and a finite-dimensional projective subspace. It is natural to assume that in this way we can find an answer to our problem, which consists just in the construction of a finite-dimensional object. We will verify this conjecture and analyse the details of the SBS-construction in this context. First of all, given a holomorphic section $\alpha \in H^0(X,L)$, the form $\rho(\alpha)$ representing it with respect to the distinguished gauge $a_I \in \mathcal{O}_{\omega}$ must have type $(1,0)$. In fact,
$$
\begin{equation*}
\rho(\alpha)=\frac{\nabla_{a_I}\alpha}{\alpha}= \frac{\partial_{a_I}\alpha+\overline \partial_{a_I} \alpha}{\alpha}= \frac{\partial_{a_I}\alpha}{\alpha} \in \Omega^{1,0}_{X \setminus D_{\alpha}}.
\end{equation*}
\notag
$$
Then by Riemann’s identity
$$
\begin{equation}
\rho(\alpha)={\rm d}\log|\alpha|_h-\imath I({\rm d}\log|\alpha|_h);
\end{equation}
\tag{16}
$$
so $S \subset X \setminus D_{\alpha}$ is an $\alpha$-SBS-submanifol if and only if $I({\rm d}\Psi_{\alpha})\big|_S \equiv 0$. Since the complex and symplectic structures are compatible, the inversion $\omega^{-1}\bigl(I({\rm d} f)\bigr)$ is equal to $ \operatorname{grad} f$, and since $S$ is Lagrangian, the $\alpha$-SBS-condition takes the simple form
$$
\begin{equation*}
\operatorname{grad}\Psi_{\alpha}(p) \in T_p S\quad\text{at each point}\ p \in S.
\end{equation*}
\notag
$$
Geometrically, this means that the Lagrangian submanifold $S$ is stationary with respect to the flow generated by the gradient field of the Kähler potential:
$$
\begin{equation}
\phi^t_{\operatorname{grad}\Psi_{\alpha}}(S) \equiv S.
\end{equation}
\tag{17}
$$
By assumption $S$ is compact, so we obtain the following geometrically transparent reformulation of the $\alpha$-SBS-condition. Given a holomorphic section $\alpha \kern-1pt \in \kern-1pt H^0(X,L)$, consider the corresponding Kähler potential $\Psi_{\alpha}$. This is a smooth real function tending to infinity at the zero divisor $D_{\alpha} \subset X$. This function has critical points (and, in the special case, critical subsets) with finite critical values in the domain $X \setminus D_{\alpha}$. The set of trajectories of the gradient flow contains the subset of trajectories of finite type connecting critical points with finite critical values (strictly speaking, each critical point is itself a zero-dimensional trajectory). Let $W(M \setminus D_{\alpha}) \subset X \setminus D_{\alpha}$ denote the subset of all points $X \setminus D_{\alpha}$ that lie on finite trajectories; by construction it is a CW-complex. Since the defining function $\Psi_{\alpha}$ is a Kähler potential, for each finite critical point the incoming subspace must be isotropic; since the vector field $\operatorname{grad}\Psi_{\alpha}$ is Liouville, the gradient flow preserves isotropy, so the whole of $W(X \setminus D_{\alpha})$ is isotropic at each smooth point. In the theory of Weinstein manifolds the CW-complex $W(X \setminus D_{\alpha})$ is called the Weinstein skeleton of $X \setminus D_{\alpha}$: in the general setting the Lyapunov function plays the role of a Kähler potential $\Psi_{\alpha}$, and in our simply connected case this function is uniquely determined by $D_{\alpha}$ (see [12]). Hence we see that in the algebro-geometric case our SBS-condition has a remarkable simple form in the language of Weinstein’s theory. Proposition 4. A Lagrangian submanifold $S \subset X \setminus D_{\alpha}$ satisfies the $\alpha$-SBS-condition if and only if $S$ lies in the Wiensetin skeleton $W(X\setminus D_{\alpha})$ of the complement to the zero divisor of the holomorphic section $\alpha \in H^0(X,L)$. In fact, $W(X \setminus D_{\alpha})$ can have smooth components of dimension at most $n$ in view of isotropy, which proves Proposition 4 (for details, see the proof of Proposition 3 in [7]). Thus, for BS-Lagrangian submanifolds, in the holomorphic case the condition of being special is closely related to the properties of the gradient flows of Kähler potentials on the complements to ample divisors in algebraic varieties. On the other hand it follows directly from the definition of the Weinstein skeleton $W(X\setminus D_{\alpha})$ that the complement $X \setminus D_{\alpha}$ is homotopic to $W(X \setminus D_{\alpha})$. In fact, the first subset is distinct from the second by the presence of half-infinite gradient trajectories, and we obtain the required homotopy by contracting these trajectories. One immediate consequence is a necessary condition for the existence of $\alpha$-SBS-Lagrangian submanifolds. Proposition 5. If $H_n(X \setminus D_{\alpha},\mathbb{Z})=0$, then the set of $\alpha$-SBS-Lagrangian submanifolds of any topological type is empty. For a proof, assume the converse: suppose there exists a smooth compact orientable Lagrangian submanifold $S \subset X \setminus D_{\alpha}$ satisfying the $\alpha$-SBS-condition. Since $n$ is the top possible dimension of components of the CW-complex $W(X \setminus D_{\alpha})$, and such $S$ must lie in $W(X \setminus D_{\alpha})$ by Proposition 4, this submanifold is a closed cycle, which cannot be a boundary for reasons of dimension, so that it represents a non-trivial homology class in $H_n(X \setminus D_{\alpha},\mathbb{Z})$, in contradiction to the hypotheses. Here is a simple example. We take as $X$ the projective space $\mathbb{C} \mathbb{P}^n$ with ample bundle $\mathcal{O}(1)$. Then for each holomorphic section $\alpha \in H^0(\mathbb{C}\mathbb{P}^n,\mathcal{O}(1))$ its zero divisor is a hyperplane $H_{\alpha} \subset \mathbb{C}\mathbb{P}^n$, and the homotopy type of the complement $X \setminus D_{\alpha}$ is that of $\mathbb{C}^n$, so it follows directly from Proposition 5 that there can be no $\alpha$-SBS-Lagrangian submanfiolds of any topological type whatsoever. Remark 3. Now it is worthwhile to return to the general situation and ask if there is a similar result for a compact simply connected symplectic manifold $(M,\omega)$ with prequantization data $(L,a)$ for a general smooth section $\alpha \in \Gamma (M,L)$. When there exists a Weinstein structure on $M \setminus D_{\alpha}$, this is Eliashberg’s well-known conjecture on exact Lagrangian submanifolds (see [12]). In our context this means that the Liouville field $\lambda_{\alpha}=\omega^{-1}(\operatorname{Im}\rho(\alpha))$ induced by $\alpha$ is gradient-like for some convex function $\Psi_{\alpha}$ (so that the pair $\Psi_{\alpha},\lambda_{\alpha})$ defines a Weinstein structure on $M \setminus D_{\alpha}$: see [12]); in addition, there exists a section $\alpha'$ proportional to $\alpha$ such that $S$ is an $\alpha'$-SBS-Lagrangian submanifold. In this case the first Eilashberg conjecture states that $S$ must represent a non-trivial class in $H_n(M \setminus D_{\alpha},\mathbb{Z})$. We stress that Proposition 5 gives no answer even in the holomorphic case because in it $\alpha$ and $\alpha'$ coincide. However, as we saw above (see Proposition 1), for a Lagrangian submanifold $S$ the SBS-condition with respect to the section $\alpha$ is related to $D$-exactness with respect to the set of zeros $D_{\alpha}$, so the following question is a natural generalization of the first Eliashberg conjecture. Question 1. Let the smooth orientable Lagrangian submanifold $S$ in a compact simply connected symplectic manifold $(M,\omega)$ be $D$-exact with respect to a $(2n- 2)$-dimensional submanifold $D \subset M$, which represents a class $\operatorname{P.D.}[\omega]\in H_{2n-2}(M,\mathbb{Z})$. Must the class $[S]$ be non-trivial in $H_n(M \setminus D_{\alpha},\mathbb{Z})$ in the general case? If not, then is it true in the case when $D$ is symplectically immersed? We look closer at the property of $D$-exactness and observations following from it in the next section. The description of SBS-Lagrangian submanifold in terms of gradient flows generated by Kähler potentials enables us to produce several simple examples (for details, see [7]). We call some of these ‘positive’ examples because they are in conformity with the above thesis (see (15)). Again, as $X$ we consider the projective space $\mathbb{C}\mathbb{P}^n$ with the standard Fubini–Study Kähler metric, but as an ample bundle we take $L=\mathcal{O}(2)$. We know from the standard course of topology that the complement $\mathbb{C} \mathbb{P}^n \setminus Q$ to a smooth quadric $Q$ is isotopic to $\mathbb{R} \mathbb{P}^n$. Consider the smooth quadric and note that $Q_0$ is real but has no real points. We look at the corresponding holomorphic section $\alpha_0 \in H^0(\mathbb{C}\mathbb{P}^n,\mathcal{O}(2))$ and examine the critical points of the corresponding Kähler potential $\Psi(\alpha_0)$. To do this we go over to the affine system of coordinates $(Z_0,\dots,Z_n)$ in $\mathbb{C}^{n+1}$ and consider conditional extrema of the function $\biggl|\,\displaystyle\sum_{i=0}^n z_i^2\biggr|^2$ under the condition $\displaystyle\sum_{i=0}^n|z_i|^2=1$. As usual, we define the auxiliary function
$$
\begin{equation*}
F_{\lambda}=\biggl|\,\sum_{i=0}^n z_i^2\biggr|^2- \lambda\biggl(\,\sum_{i=0}^n z_i \overline z_i-1\biggr),
\end{equation*}
\notag
$$
solve the system of equations
$$
\begin{equation}
\frac{\partial F_{\lambda}}{\partial z_i}=0, \quad \frac{\partial F_{\lambda}}{\partial \overline z_i}=0, \qquad i=0,\dots,n,
\end{equation}
\tag{18}
$$
discard the solutions with trivial $\lambda$, which correspond to infinite growth along the divisor $Q_0$, and take the quotient by the phase rotations, descending to the base of the Hopf fibration. As a result, we obtain a whole critical set $S_0 \subset \mathbb{C} \mathbb{P}^n$, which is isomorphic to $\mathbb{R} \mathbb{P}^n$. Of course, the latter manifold is not always orientable, but when $n$ is odd, it is well suited as an example of an SBS-Lagrangian submanifold with respect to the holomorphic section $\alpha_0 \in H^0(\mathbb{C}\mathbb{P}^n,\mathcal{O}(2))$. Thus, comparing the results obtained for the same algebraic variety $\mathbb{C}\mathbb{P}^n$, but different bundles we arrive at essentially different answers, although from the point of view of the general classification of Lagrangian submanifolds the conditions defining Lagrangian manifolds with respect to the the symplectic forms $\omega_{\rm FS}$ and $2\omega_{\rm FS}$ are equivalent. The calculations of local extrema in the previous example look rather naive, but the analysis of the gradient flows of Kähler potentials on projective varieties does not fit in the frameworks of sophisticated theories even in simplest cases. For instance, let us consider a construction from the theory of algebraic curves, one of the best developed areas in algebraic geometry. Consider an algebraic curve $C$ of genus greater than one and represent it, in the framework of differential geometry, as a Riemann surface $\Sigma$ with complex structure $I$ (note that in the two-dimensional case each complex structure is integrable). According to the general theory, there exists a unique, up to a positive real constant, uniformizing metric $G$ compatible with $I$, and we can choose the constant so that the corresponding Kähler form $\Omega$ satisfies $\displaystyle\int_{\Sigma}\Omega=2g-2$. Thus, there exists a well-defined Hermitian triple $(G,I,\Omega)$, which defines an Hermitian structure $h$ on the contangent bundle. Therefore, for each section $\alpha \in \Gamma(\Sigma,T^*\Sigma)$ — for instance, for an arbitrary holomorphic differential in $H^0(\Sigma,K_{\Sigma})$ — the corresponding function is well defined by If $\alpha$ is a holomorphic differential, then $\Psi(\alpha)$ has poles in the zero divisor $D_{\alpha} \subset \Sigma$, which consists of $2g-2$ points in the general case. By the theory of subharmonic functions other critical points can have Morse index at most 1, so they are minima or saddle points. Furthermore, for a general section the number of such finite critical points is finite. Consider the graph $\Gamma(\alpha) \subset \Sigma \setminus D_{\alpha}$ formed by the finite trajectories of the gradient flow of $\Psi(\alpha)$. By construction its edges are oriented, so the complex-valued function
$$
\begin{equation}
\mathcal{W}(I,\alpha)=\sum_{e_j \subset \Gamma(\alpha)}\int_{e_j}\alpha \in \mathbb{C}
\end{equation}
\tag{19}
$$
is well defined, where the sum is taken over all oriented edges of $\Gamma(\alpha)$. (We stress that the holomorphic differential is involved twofold in the definition of $\mathcal{W}$: first it defines the Weinstein skeleton $\Gamma(\alpha)$, and then we integrate the same differential along oriented edges.) If the section $\alpha$ is not general and $\Psi(\alpha)$ can have critical sets, then, as such sets are non-oriented closed one-dimensional curves, it can merely be excluded from integration in a natural way. Now, by definition $\mathcal{W}(I,\alpha)$ can be extended to the total space of the bundle $\pi\colon\mathcal{H} \to \mathcal{M}_g$, where $\mathcal{M}_g$ is the moduli space of algebraic curves of genus $g$, and a fibre in $\mathcal{H}$ is the space of holomorphic differentials $H^0(\Sigma_I,K_{\Sigma})$. The following question arises here naturally. Question 2. What are the properties of the function $\mathcal{W}\colon \operatorname{tot}\mathcal{H} \to \mathbb{C}$ defined by (19)? Clearly, answering this question is an exercise in an analysis of the gradient flows of Kähler potentials in the lowest dimension (for details, see [7], § 2, remarks on the non-simply connected case after Theorem 1). Here are a few other observations opening way to the construction of SBS-Lagrangian submanifolds which extend the series of ‘positive’ examples from the standpoint of our central thesis. From Proposition 4 in [7] it follows that the SBS-condition on a Lagrangian submanifold $S$ is stable under the transition from a section $\alpha_0$ to another section $\alpha_1$, provided that $(\alpha_1/\alpha_0)\big|_S \in C^{\infty}(S,\mathbb{R}_+)$. In fact, this is an immediate consequence of Definition 1: if the ratio of $\alpha_0\big|_S$ to $\sigma_S$ preserves its phase, then, since the quotient $(\alpha_1/\alpha_0)\big|_S$ is real, the ratio of $\alpha_1\big|_S$ to $\sigma_S$ also preserves phase. For example, this shows that the Lagrangian submanifold $\mathbb{R} \mathbb{P}^n \subset \mathbb{C} \mathbb{P}^n$ is SBS not only with respect to any section with zeros on but the same holds for any real quadric
$$
\begin{equation*}
Q=\biggl\{\,\displaystyle\sum_{i=0}^n a_i z_i^2=0\biggr\},\qquad a_i \in \mathbb{R}_+.
\end{equation*}
\notag
$$
Thus, we can avoid cumbersome calculations for a whole class of holomorphic sections and obtain results on smooth components of Weinstein skeletons from the basic definitions in special Bohr–Sommerfeld geometry. Another observation, which is Proposition 5 in [7], allows us to perform a reduction of dimension: assume that in the ambient algebraic variety $X$ we have an $\alpha$-SBS-Lagrangian submanifold $S$. If for a smooth algebraic subvariety $Y \subset X$ we have $\dim_{\mathbb{R}}(Y \cap S)=\dim_{\mathbb{C}}Y$ and the intersection $Y \cap S$ is smooth, then it is an $\alpha'$-SBS-Lagrangian submanifold of $Y$ with respect to the section $\alpha'=\alpha\big|_Y$. For example (see [7], Example 4), we can construct a Lagrangian torus in the 2-quadric $Q=\{z_0z_1=z_2 z_4\} \subset \mathbb{C} \mathbb{P}^3$ which satisfies the SBS-condition with respect to the intersection of $Q$ with the quadric $Q_0$ from the above examples. The most interesting ‘positive’ example was presented in [7] as Example 5: for it we must combine all observations used above for the construction of SBS-Lagrangian submanifolds. Namely, consider the variety $F^3$ of complete flags in $\mathbb{C}^3$, which can ve realized by the incidence cycle
$$
\begin{equation*}
\mathcal{U}=\biggl\{\,\sum_{i=0}^2 x_i y_i=0\biggr\} \subset \mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2,
\end{equation*}
\notag
$$
where $[x_i]$ and $[y_j]$ are the homogeneous coordinates on the first and second projective planes, respectively. We construct an SBS-Lagrangian submanifold for a section of the bundle $\mathcal{O}(1,1)\big|_{\mathcal{U}}$ whose zero divisor $D_0$ has the equation To do this, first, using calculations of condition extrema, we find an SBS-Lagrangian submanifold in the Cartesian product $\mathbb{C}\mathbb{P}^2 \times \mathbb{C}\mathbb{P}^2$, namely, the submanifold
$$
\begin{equation}
S_0=\{[x_0:x_1:x_2] \times [\overline x_0:\overline x_1:-\overline x_2]\} \subset \mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2.
\end{equation}
\tag{20}
$$
After that, considering the intersection of $S_0$ and $\mathcal{U}$ we obtain a submanifold $S \subset \mathcal{U}$, which, first, has the required dimension of three and, second, is topologically isomorphic to a 3-sphere. This Lagrangian sphere turns out to be Hamiltonially equivalent to the well-known Gelfand–Tsetlin sphere. In the next section we discuss thoroughly why $H_3(\mathcal{U} \setminus D, \mathbb{Z})=\mathbb{Z}$, which shows that $W(\mathcal{U} \setminus D)$ has no other components. The divisor
$$
\begin{equation*}
D_0=\{x_0 y_0+x_1 y_1-x_2 y_2=0\} \subset \mathcal{U}
\end{equation*}
\notag
$$
is not a smooth subvariety, but admits variations of the form so that, first, for real positive $A$, $B$, and $C$ the relation $(\alpha_1/\alpha_0)\big|_{S_0}$ is real and, second, for general $A,B,C \in \mathbb{R}_+$ the corresponding divisor $D_1$ is smooth. Thus, the Lagrangian sphere $S$ is also SBS with respect to the section $\alpha_1$. However, we will see in the next section that for a smooth divisor $D_1$ the group $H_n(\mathcal{U}\setminus D_1,\mathbb{Z})$ is no longer equal to $\mathbb{Z}[S]$, so the Weinstein skeleton $W(\mathcal{U} \setminus D_1)$ is not exhausted by the sphere $S$, its smooth component, but has a more complicated structure. That is, in spite of the simple construction of the smooth component $S$, the full answer is more complex. Now, from separate examples when a divisor is associated with a Lagrangian submanifold, we go over to the construction of the moduli space hypothesized in our thesis. Consider the simplest case: $X=\mathbb{C} \mathbb{P}^1$ and the bundle $L$ is $\mathcal{O}(2)$. As we saw above, for real non-degenerate ‘quadrics’ $D=\{p,\overline p\} \subset \mathbb{C}\mathbb{P}^1$ formed of pairs of distinct conjugate points, the corresponding SBS-Lagrangian submanifold is always the smooth circle $\mathbb{R} \mathbb{P}^1 \subset \mathbb{C} \mathbb{P}^1$, because in this case $H_1(X \setminus D,\mathbb{Z})$ has dimension one. Using the group of Kähler isometries of the Fubini–Study metric we can extend this result to any non-degenerate ‘quadric’. Furthermore, in the case of a multiple section the set of SBS-Lagrangian submanifolds is empty. Hence we obtain an answer in the form $\mathcal{M}_{\rm SBS}=\mathbb{C} \mathbb{P}^2 \setminus \Delta$, where $\mathbb{C} \mathbb{P}^2=|\mathcal{O}(2)|$ is a complete linear system of divisors of degree 2, and $\Delta$ is the discriminant curve corresponding to multiple zeros. In this simplest case we obtain a remarkable answer: the required moduli space is also an algebraic variety. Making a step further, over the same base we consider the bundle $L=\mathcal{O}(3)$. For a general section we have $H_1(X \setminus D,\mathbb{Z})=\mathbb{Z} \oplus \mathbb{Z}$, and by analogy with the previous case it is natural to assume that, as SBS-Lagrangian submanifolds, we obtain three smooth circles. However, nothing of the sort occurs even for ‘nice’ sections. For example, in the case of a section with divisor of zeros $D=\{z_0^3-z_1^2=0\}$ the corresponding Weinstein skeleton $W(X \setminus D)$ has the following form: two points $m_1,m_2 \in \mathbb{C} \mathbb{P}^1$ correspond to the two minima, and three points $s_1,s_2,s_3 \in \mathbb{C}\mathbb{P}^1$ correspond to the three saddle points of the corresponding Kähler potential. The minima $p_1$ and $p_2$ are connected by three arcs $\gamma_i$, going through the $s_i$, which are the separatrices of the gradient flow that separates the half-infinite trajectories starting at $p_1$ and $p_2$ and going to infinity at the three zeros of the section. It is easy to see from symmetry that any two curves $\gamma_i$ and $\gamma_j$ meet at $p_1$ and $p_2$ under an angle of $2\pi/3$. Hence in this simple case $W(X \setminus D)$ admits no smooth SBS-Lagrangian submanifolds! It is easy to see that the same holds for an arbitrary smooth section. For some sections one loop in the Weinstein skeleton is smoothened, but such sections are distinguished by very complicated non-algebraic conditions (for details of calculations, see [9]). The reason is that, as shown by Eliashberg with co-authors (see [12]), the Weinstein skeleton $W(X \setminus D)$ of the complement to an ample divisor is always very singular. In the best case we can expect that the Weinstein skeleton has tree singularities but no smooth components at all. Thus, the straightforward definition of the moduli space of SBS-Lagrangian submanifolds does not actually work. However, this is not that discouraging: the above definitions and constructions in connection with Weinstein’s theory suggest an alternative way to the construction of appropriate finite-dimensional objects in terms of the Lagrangian geometry of algebraic varieties, which we discuss in the next section. What is important, SBS-geometric considerations for algebraic varieties have already taken us to a remarkable manifestation of the duality between complex and symplectic geometry: with an ample divisor $D \subset X$ we associate in a natural way a combinatorial set of Lagrangian or, more generally, isotropic objects, once a suitable Hermitian structure has been fixed on the relevant holomorphic linear bundle. Similarly to how real-life objects cast shadows when the sun rises, these shadows change with the position of the sun, and can be quite non-smooth and complicated, the ‘shadows’ cast by ample divisors are deformed under variations of the Hermitian structure preserving their homotopy types, because the Weinstein skeleton $W(X \setminus D)$ is always homotopic to the complement $X \setminus D$ itself. Moreover, in the simplest cases we clearly see how the algebraic variety $X$ is realized as some ‘envelope’ over the $\alpha$-SBS-Lagrangian submanifold $S$ and the corresponding divisor $D_{\alpha}$, provided that this $S$ exists indeed. 3. Antithesis: $D$-exact Lagrangian submanifoldsOne of the central ideas of Weinstein’s theory consists in considering, in place of non-smooth components of the Weinstein skeleton $W(X \setminus D)$, smooth exact Lagrangian submanifolds of the complement $X \setminus D$: Eliashberg’s famous conjectures suggest that such submanifolds carry the same information as components of the skeleton (see [12]). However, an important distinction is that components of the skeleton are rigid, whereas a smooth exact Lagrangian submanifold has a continuous deformation space. Thus, it is natural to assign to components of $W(X \setminus D)$ classes of equivalent exact Lagrangian submanifolds of $X \setminus D$. We return to the last example in the previous section: let $X=\mathbb{C} \mathbb{P}^1$ and $L= \mathcal{O}(3)$. For a general section the zero divisor consists of three different points:
$$
\begin{equation*}
D_3=\{p_1,p_2,p_3\},\qquad p_i \in \mathbb{C} \mathbb{P}^1,
\end{equation*}
\notag
$$
and a smooth loop $\gamma \subset \mathbb{C} \mathbb{P}^1 \setminus D_3$ is $D$-exact with respect to $D_3$ if the following condition is satisfied: the complement $\mathbb{C} \mathbb{P}^1 \setminus \gamma$ falls into a union of two discs $B_1 \cup B_2$ such that $B_l$ contains $l$ points from the set $\{p_i\}$ and
$$
\begin{equation*}
\int_{B_2}\omega_{\rm FS}=2\int_{B_1}\omega_{\rm FS}.
\end{equation*}
\notag
$$
It is easy to see that each such loop $\gamma$ can be constructed as follows: we select one point $p_i$ from the given three, draw a small circle $\gamma_{\varepsilon}$ around it and then ‘blow up’ this circle by considering a family $\gamma_t$, $t \in [\varepsilon,1/3]$, such that the symplectic area of the disc bounded by $\gamma_t$ is $t$ but $\gamma_t \cap D_3=\varnothing$ for each $t$. Using this procedure, to each $D$-exact smooth loop we can assign the corresponding point $p_i$; on the other hand, modulo Hamiltonian isotopies of the complement $\operatorname{Ham}(\mathbb{C}\mathbb{P}^1 \setminus D_3)$ this point defines uniquely the corresponding class of equivalent $D$-exact loops. In fact, the only invariant of Hamiltonian isotopies of a smooth loop on a sphere is the symplectic area of the disc bounded by this loop, and by considering Hamiltonian isotopies on the complement $\mathbb{C}\mathbb{P}^1 \setminus D_3$ we ensure that the smooth loop cannot ‘leap’ over the point $p_i \in D_3$. Thus, for a fixed general divisor $D_3 \in |\mathcal{O}(3)|$ we see that the quotient space (smooth $D$-exact loops)/(Hamiltonian isotopies) is naturally isomorphic to the set of three simple zeros $\{p_1,p_2,p_3\}$ of the corresponding section. Next we look at a non-general case: let the divisor $D_2 \in |\mathcal{O}(3)|$ have a zero of multiplicity two at a point $p_2$. Then it is easy to see that the above procedure of the construction of a $D$-exact loop $\gamma_{1/3}$ is valid for the point $p_1$ but the loop we obtain can also be constructed as $\gamma_{2/3}$, by starting from $p_2$ and modifying the procedure for the case of a point with multiplicity. Thus, for $D_2$ the same quotient space is precisely isomorphic to the set of simple zeros of the corresponding section. Now, in the even more special situation of one zero $p_3$ of multiplicity three there exist no $D$-exact smooth loops: each smooth loop $\gamma \subset \mathbb{C} \mathbb{P}^1 \setminus p_3$ bounds a disc with positive symplectic area and trivial intersection with $D_1$, in contradiction to Definition 2. Thus, in this case the quotient space is also isomorphic to the (now empty) set of simple zeros. Summarizing: assigning to divisors $D \in |\mathcal{O}(3)|$ the quotient spaces (smooth $D$-exact loops)/(Hamiltonian isotopis) induces a non-ramified covering over the projective space $|\mathcal{O}(3)|$, which can be described as follows. Consider the following subvariety of bidegree $(1,3)$ of a Cartesian product of projective spaces:
$$
\begin{equation}
Y=\{y_0 z_0^3+y_1 z_0^2 z_1+y_2 z_0 z_1^2+y_3 z_1^3=0\} \subset |\mathcal{O}(3)| \times \mathbb{C} \mathbb{P}^1,
\end{equation}
\tag{21}
$$
with the canonical projection $q_1\colon Y \to |\mathcal{O}(3)|$ onto the first factor, which distinguishes the ramification divisor $\Delta \subset Y$. It is easy to see that the space under consideration, of pairs of the form $(D,\langle S\rangle)$, where $D \in |\mathcal{O}(3)|$ and $\langle S\rangle$ is the class of equivalent $D$-exact smooth loops modulo Hamiltonian isotopies of the complement $\mathbb{C} \mathbb{P}^1 \setminus D$, is naturally isomorphic to the affine algebraic variety $Y \setminus \Delta$. This answer exceeds our expectations: we have not only defined a smooth moduli space with elements described in terms of Lagrangian geometry, but this variety also admits an algebraic structure. Moreover, it specifies a pair of precisely the form with which we started our considerations, namely, an algebraic variety and a very ample divisor, so that we can repeat our construction for $Y$ and the complete linear system $|\Delta|$. On the other hand we can view the above arguments as a generalization of the SBS-construction, excluding from considerations the gauge parameter, prequantization connection $a$. In fact, Proposition 1 in this paper establishes a correspondence between the pairs $(D_{\alpha},S)$, where $S$ is a $D$-exact Lagrangian submanifold of the complement $M \setminus D_{\alpha}$, and the elements of the incidence cycle $\mathcal{U}_{\rm SBS}(a)$ for a certain suitable prequantization connection. The quotient correspondence $\mathcal{F}$ in (9) ensures that the moduli space of $D$-exact Lagrangian loops in $\mathbb{C}\mathbb{P}^1$ with respect to the polarization $L=\mathcal{O}(3)$, which we constructed above, is a natural re-formulation in a more invariant form of the central idea of our thesis (see (15)). We discuss this link in greater detail in the last section, and devote the rest of this section to the definition, main properties, and a more interesting example (than the one above) of a moduli space of $D$-exact Lagrangian submanifolds. The source data for our construction are just as before: a projective variety $X$ with an ample bundle $L$. Fixing an appropriate Hermitian structure $h$ on $L$ we consider the corresponding symplectic form $\omega$. Fix topological data required in Bohr–Sommerfeld geometry: a class $[S] \in H_n(X,\mathbb{Z})$ and a topological type $\operatorname{top}S$. Given a divisor $D \in |L|$, consider the space $\mathcal{H}(D)$ of smooth $D$-exact (with respect to this $D$) Lagrangian submanifolds of the complement $X \setminus D$ which have the topological type fixed above. Note that on the space $\mathcal{H}(D)$ we have a natural action of the group $\operatorname{Ham}(X \setminus D)$ of Hamiltonian isotopies of $X \setminus D$; moreover, $\mathcal{H}(D)$ is invariant under this action. In fact, under any Hamiltonian isotopy in $\operatorname{Ham}(X \setminus D)$ a Lagrangian submanifold remains within $X \setminus D$. On the other hand, being a Hamiltonian isotopy it preserves the Bohr–Sommerfeld condition, which ensures the invariance of the space $\mathcal{H}(D)$ (for details, see [9]). Hence we can consider the quotient space $\mathcal{H}(D)/\operatorname{Ham}(X \setminus D)$, and the main property of such spaces is described in the following statement. Proposition 6. The quotient space $\mathcal{H}(D)/\operatorname{Ham} (X \setminus D)$ is a discrete set. In fact, Proposition 4 in [9] shows that, given a $D$-exact Lagrangian submanifold $S_0$, any other $D$-exact Lagrangian submanifold in its Darboux–Weinstein neighbourhood $\mathcal{O}_{\rm DW}(S_0)$ must be a Hamiltonian deformation of $S_0$, and therefore we can choose representatives $S_1,\dots,S_k,\dots$ of elements of $\mathcal{H}(D)/\operatorname{Ham}(X \setminus D)$ so that no $S_i$ lies fully in any neighbourhood $\mathcal{O}_{\rm DW}(S_j)$. The correspondence $|L| \ni D \mapsto \mathcal{H}(D)/\operatorname{Ham}(X \setminus D)$ globalizes to a covering
$$
\begin{equation}
\pi\colon \widetilde{\mathcal{M}}_{\rm SBS} \to |L|, \qquad \pi^{-1}(D)=\mathcal{H}(D)/\operatorname{Ham}(X \setminus D),
\end{equation}
\tag{22}
$$
where the space $\widetilde{\mathcal{M}}_{\rm SBS}$ consists of the pairs $(D,\langle S\rangle)$ whose second component is a class of $D$-exact (with respect to $D$ in question) Lagrangian submanifolds of fixed topological type. One remarkable consequence of $D$-exactness is double stability, both with respect to small deformations of the divisor $D$ in the complete linear system $|L|$ and with respect to small Hamiltonian deformations of the Lagrangian submanifold $S$. Namely, the following result holds. Proposition 7. (1) Let $D_t \in |L|$ be a family of divisors such that for $t \in [0,T]$ the intersection $D_t \cap S$ is empty. If $S$ is $D$-exact with respect to some $D_0$, then it is $D$-exact with respect to each $D_t$. (2) Let $S_t$ be a small Hamiltonian deformation of a Lagrangian submanifold $S_0 \subset X \setminus D$ such that for each $t \in [0,T]$ the intersection $S_t \cap D$ is empty. If $S_0$ is $D$-exact with respect to $D$, then each $S_t$ is also $D$-exact. In fact, a small deformation does not change the intersection index $\operatorname{ind}(B\cap D_t)$ until $D_t$ intersects $S$, which immediately establishes the first type of stability. On the other hand, under a Hamiltonian deformation the $S_t$ remain BS, so the symplectic area $\displaystyle\int_B \omega$ does not change, nor does the intersection index $\operatorname{ind}(B \cap D)$, so by Definition 2 in § 1 the property of $D$-exactness is stable under such deformations. Thus, we can naturally introduce a neighbourhood system in $\widetilde{\mathcal{M}}_{\rm SBS}$ by the following rule. Given a point $(D_0,\langle S\rangle) \in \widetilde{\mathcal{M}}_{\rm SBS}$, we choose a representative of the class $\langle S \rangle$, which is a $D$-exact Lagrangian submanifold $S \subset X \setminus D$. Then in the projective space $|L|$ the point $D_0$ has a neighbourhood $\mathcal{O}_{\varepsilon} (D_0) \subset |L|$ such that for each divisor $D \in \mathcal{O}_{\varepsilon}(D_0)$ we have $D \cap S=\varnothing$. The corresponding neighbourhood in $\widetilde{\mathcal{M}}_{\rm SBS}$ if the set of pairs of the form $(D,\langle S \rangle_D)$, where the equivalence class $\langle S \rangle_D$ is represented by the same Lagrangian submanifold $S$, but modulo $\operatorname{Ham}(X \setminus D)$, rather than modulo the action of $\operatorname{Ham}(X \setminus D_0)$. However, the slice of group action that we consider models the whole quotient space, so we obtain a description of the corresponding neighbourhood in $\widetilde{\mathcal{M}}_{\rm SBS}$. It is easy to see that the whole of $\widetilde{\mathcal{M}}_{\rm SBS}$ acquires the induced smooth structure. Moreover, the Kähler structure is pulled back from $|L|$ in the natural way, which shows that this manifold admits a Kähler structure (for details, see [9]). In the example presented at the beginning of this section such a moduli space $\widetilde{\mathcal{M}}_{\rm SBS}$ for $X=\mathbb{C}\mathbb{P}^1$ and $L=\mathcal{O}(3)$, with the only possible choice of topological data $[S]=0$ and $\operatorname{top} S \equiv S^1$, turns out to be isomorphic to the affine algebraic variety $Y \setminus \Delta$, where $Y$ is the projective variety explicitly defined by (21) and $\Delta$ is an ample divisor. In would be fairly naive to believe that such a property holds in the general case, as the complexity of the problem increases indefinitely. For example, in the case of the complement $\mathbb{C} \mathbb{P}^1 \setminus \{p_1,p_2\}$ a $D$-exact loop must represent a non-trivial class in the group $H_1(X \setminus D,\mathbb{Z})$, while in the general case this is one of Eliashberg’s basic conjectures, which are open so far (see [12]). On the other hand finding all possible classes in the quotient space $\mathcal{H}(D)/ \operatorname{Ham}(X \setminus D)$ for a fixed divisor $D$ is an extremely difficult problem: having found some set of classes $\langle S_1 \rangle,\dots,\langle S_k \rangle$, there is no standard way to show that there exist no more of them. However, in the construction of $\widetilde{\mathcal{M}}_{\rm SBS}$, as the first step we must introduce an appropriate Hermitian structure $h$ on the ample bundle $L$, and so for each divisor $D \in |D|$, in the presence of an Hermitian structure we can define the Weinstein skeleton $W(X \setminus D)$, which can naturally be involved in our considerations of this section. Namely, we give the following definition. Definition 3. A class $\langle S \rangle \in \mathcal{H}(D)/\operatorname{Ham}(X \setminus D)$ is said to be stable if it has a representative $S_1 \subset X \setminus D$ included in a homotopy $S_t$, $t \in [0,1]$, which is smooth for $t \in (0,1]$ and such that $S_0 \subset W(X \setminus D)$ is a closed cycle in the Weinstein skeleton, while each $S_t$, $t \in (0,1],$ is a smooth $D$-exact Lagrangian submanifold. In [9] we called the homotopy $S_t$ in Definition 3 a Bohr–Sommerfeld resolution of the singular cycle $S_0 \subset W(X \setminus D)$, as the family $S_t$ is in fact a kind of a resolution. It follows directly from Definition 3 that each representative of a stable class $\langle S \rangle$ must represent a non-trivial holomology class in $H_n(X \setminus D, \mathbb{Z})$. In fact, a closed cycle $S_0$ has dimension $n$, which is the highest possible dimension for the CW-complex $W(X \setminus D)$, so it cannot represent a trivial class, and the same holds for its resolutions $S_t$. On the other hand it was proved in [9] (Proposition 6) that the closed cycle $S_0 \subset W(X \setminus D)$ can give rise to at most one class of Hamiltonian equivalent $D$-exact Lagrangian submanifolds of fixed topological type. Then we can give the following definition. Definition 4. The stable component $\mathcal{M}^{\rm st}_{\rm SBS} \subset \widetilde{\mathcal{M}}_{\rm SBS}$ is the subset of pairs $(D,\langle S \rangle) \in \widetilde{\mathcal{M}}_{\rm SBS}$ such that $\langle S \rangle$ is a stable class. Clearly, the stable component inherits the structure of a covering
$$
\begin{equation}
\pi_{\rm st}\colon \mathcal{M}^{\rm st}_{\rm SBS} \to |L|,
\end{equation}
\tag{23}
$$
and for a fixed topological type $([S] \in H_n(X,\mathbb{Z}),\operatorname{top}S)$ this is a finite covering. However, to carry over Proposition 7 to the case of the stable component $\mathcal{M}_{\rm SBS}^{\rm st}$ requires an additional analysis because the group $H_n(X \setminus D,\mathbb{Z})$ depends on the degeneracy of the divisor $D$. In the complete linear system $|L|$ a general element is represented by a smooth algebraic subvariety of codimension 1, but there exists a stratification $|L|=K_0 \cup K_1 \cup \dots \cup K_m$ by the degeneracy degrees of divisors. For example, in the case when $X=\mathbb{C}\mathbb{P}^1$ and $L=\mathcal{O}(3)$, as in the beginning of this section, $K_1$ corresponds to the presence of multiple zeros and $K_2$ to multiplicity three. Furthermore, the group $H_n(X \setminus D,\mathbb{Z})$ decreases with the increasing degeneracy of $D$: in the same example the component $K_i$ corresponds to rank $2-i$. In the general case the reduction is similar: in going over from $K_i$ to $K_{i+1}$ the rank of $H_n(X \setminus D,\mathbb{Z})$ decreases. For this reason the assertion that the smooth component $\mathcal{M}_{\rm SBS}^{\rm st}$ is smooth must be substantiated additionally in the case when $D_0$ is not a general divisor. Now consider a more complicated example from [10]: let $X$ be the complete flag variety in $\mathbb{C}^3$, which is realized by an incidence cycle $\mathcal{U}$ in the Cartesian product of projective planes $\mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2$, and let the ambient bundle $L$ be the restriction of $\mathcal{O}(1,1)$. In each plane we take the homogeneous coordinates $[x_0:x_1:x_2]$ and $[y_0: y_1: y_2]$ and consider the corresponding Kähler forms $\omega_x$ amd $\omega_y$; moreover, the variety in question has the equation
$$
\begin{equation*}
\mathcal{U}=\biggl\{\,\sum_{i=0}^2 x_i y_i=0\biggr\}.
\end{equation*}
\notag
$$
We denote the canonical projections onto direct summands by $\pi_x$ and $\pi_y$, respectively; then $L=(\pi_x^* \mathcal{O}(1) \otimes \pi_y^* \mathcal{O}(1))\big|_{\mathcal{U}}$ is a very ample bundle on $\mathcal{U}$, and as the corresponding Kähler form we take
$$
\begin{equation*}
\omega=(\pi_x^* \omega_x \oplus \pi_y^* \omega_y)\big|_{\mathcal{U}}.
\end{equation*}
\notag
$$
Holomorphic sections of $L \to \mathcal{U}$ are represented by traceless matrices: each equation of the form $\displaystyle\sum_{i,j=0}^2 a_{ij} x_i y_j=0$, corresponding to the matrix $A=\langle a_{ij}\rangle$ defines a divisor in the complete linear system $|\pi_x^* \mathcal{O}(1) \otimes \pi_y^* \mathcal{O}(1)|$. However, when restricting to the variety $\mathcal{U}$, which is also defined by a similar condition with the identity matrix $A=E$, we must take the quotient modulo addition of scalar matrices, which yields the condition on the trace. There exists a Lagrangian sphere in the flag manifold, which is called the Gelfand–Tsetlin sphere and can explicitly be defined as follows. Let $S_i \subset \mathcal{U}$ be defined by
$$
\begin{equation*}
S_i=\{y_i=-\overline x_i,\ y_j=\overline x_j,\ y_k=\overline x_k\},
\end{equation*}
\notag
$$
where $(i,j,k)$ is a permutation of the indices $0$, $1$ and $2$. It is easy to see that for each $i$ these equations define a sphere. Clearly, each Lagrangian sphere is $D$-exact with respect to any divisor disjoint from it. To construct the stable component $\mathcal{M}_{\rm SBS}^{\rm st}$ we fix topological data in the form $[S]=0$ and $\operatorname{top} S=S^3$. Below we need the following facts. First, $\mathcal{U}$, which is not toric, admits a regular action of a 2-torus which is generated by the moment maps
$$
\begin{equation}
\begin{aligned} \, F_1=\frac{|x_0|^2-|x_1|^2}{\sum_{i=0}^2 |x_i|^2}- \frac{|y_0|^2-|y_1|^2}{\sum_{i=0}^2 |y_i|^2}, \\ F_2=\frac{|x_0|^2-|x_2|^2}{\sum_{i=0}^2 |x_i|^2}- \frac{|y_0|^2-|y_2|^2}{\sum_{i=0}^2 |y_i|^2}\,. \end{aligned}
\end{equation}
\tag{24}
$$
These two functions commute with respect to the Poisson bracket $\{\,\cdot\,;\,\cdot\,\}_{\omega}$; they have precisely six common critical points of the form where the pairs $(i,j)$ and $(k,l)$ must be distinct; the Hamiltonian vector fields $X_{F_1}$ and $X_{F_2}$ define an integrable two-dimensional distribution away from the following nine rational curves: the three lines of the form the three lines of the form (here $(i,j,k)$ is a permutation of $(0,1,2)$), and the three ‘diagonal’ lines of the form The image of the map $(F_1,F_2)\colon \mathcal{U} \to \mathbb{R}^2$ is a convex hexagon $P_{\mathcal{U}}$; the preimages of its vertices are the common critical points of $F_1$ and $F_2$, the preimages of its sides are the three lines of the first type and three lines of the second, while ‘diagonal’ lines are taken to principal diagonals of $P_{\mathcal{U}}$, although they are not their full preimages. Orbits of the incomplete toric action generated by $F_1$ and $F_2$ are six points, nine families of circles lying on the the nine distinguished lines, and the complement to the nine lines is foliated by smooth 2-tori.Since we are only interested in Lagrangian submanifolds representing non-trivial classes in the homology of $\mathcal{U} \setminus D$, we find $H_3(\mathcal{U} \setminus D, \mathbb{Z})$ for all $D$. Fibres of the canonical projection $\pi_x\colon \mathcal{U} \to \mathbb{C} \mathbb{P}^2$ are projective lines. If the zero divisor $D$ of a section is described by a traceless matrix $A_D$, then for an arbitrary point $p=[x_0: x_1: x_2] \in \mathbb{C} \mathbb{P}^2$ the intersection $D \cap \pi^{-1}_x(p)$ of the divisor with a fibre is defined by the system of equations
$$
\begin{equation}
\sum_{i=0}^2 x_i y_i=0, \qquad \sum_{j=0}^2(a_{ij} x_i) y_j=0,\quad i=0,1,2.
\end{equation}
\tag{25}
$$
Its rank depends on whether $(x_0,x_1,x_2)$ is an eigenvactor of $A_D$: if it is, then this intersection contains the whole fibre, while if it is not, then the system has rank two and the intersection has codimension one. Hence we see that $\mathcal{U} \setminus D$ is homotopic to $\mathbb{C} \mathbb{P}^2$ without the points and subsets corresponding to eigenvalues of $A_D$. Again, as for the pair $(\mathbb{C} \mathbb{P}^1,\mathcal{O}(3))$, we have a stratification where $i+1$ is the maximum muliplicity of an eigenvalue of $A_D$.For $K_0$ we have the following: $\mathcal{U} \setminus D$ is homotopic to $\mathbb{C}\mathbb{P}^2 \setminus \{p_1,p_2,p_3\}$ (we contract each fibre punctured at a point and discard the whole fibres over the $p_i$), so that
$$
\begin{equation*}
H_3(\mathcal{U} \setminus D, \mathbb{Z})=H_3(\mathbb{C} \mathbb{P}^2 \setminus \{p_1,p_2,p_3\},\mathbb{Z})=\mathbb{Z} \oplus \mathbb{Z}.
\end{equation*}
\notag
$$
For $K_1$ the matrix $A_D$ has two different eigenvalues, so there exists a one- and a two-dimensional eigenspace or a pair of one-dimensional eigenspaces (the case of a Jordan block). In the first case we remove from $\mathbb{C}\mathbb{P}^2$ the lines and the point away from it that correspond to the eigenspaces. The removal of a line transforms the projective plane into the affine plane $\mathbb{C}^2$, so in this case $\mathcal{U} \setminus D$ is homotopic to $\mathbb{C}^2 \setminus p$, that is, In the second case $\mathcal{U}\setminus D$ is homotopic to $\mathbb{C}\mathbb{P}^2$ punctured at two points, so we arrive at the same answer.For $K_2$ all three eigenvalues coincide and are equal to zero and, again, two cases are possible: a Jordan block of rank 3 with a unique eigenvector or a Jordan block of rank 2. In the first case $\mathbb{C}\mathbb{P}^2$ is punctured at a point and in the second a line is removed, but in either case we obtain the trivial group $H_3(\mathcal{U} \setminus D, \mathbb{Z})$. Hence the rank of $H_3(\mathcal{U} \setminus D,\mathbb{Z})$ is equal to $2-i$ for a divisor in $K_i$. In addition, the group contains three primitive cycles for a general divisor $D \in K_0$, one cycle for $D \in K_1$, and there are no elements for $D \in K_2$. Thus, the stable component can contain at most three elements in the fibre over a general point $D \in |L|$, at most one element over $D \in K_1$, and there is nothing to do for $D \in K_2$. Note that the case of a reducible section corresponds precisely to a two-dimensional eigensubspace, and for a general reducible section we have $H_3(\mathcal{U} \setminus D, \mathbb{Z})=\mathbb{Z}$. Next we use the available symmetries and consider two model divisors invariant under the $T^2$-action described above. Consider the map
$$
\begin{equation}
\psi\colon \mathcal{U} \setminus B \to \mathbb{C}\mathbb{P}^2,
\end{equation}
\tag{26}
$$
where $B$ is the union of the six boundary lines equal to the preimages of sides of the hexagon $ P_{\mathcal{U}}$, which is defined explicitly by Then an open part of the flag variety $\mathcal{U} \setminus B$ is fibred over the projective line
$$
\begin{equation*}
L_0=\{w_0+w_1+w_2=0\} \subset \mathbb{C} \mathbb{P}^2;
\end{equation*}
\notag
$$
and each fibre $\psi^{-1}(p)$, $p \in L_0$ is compactified in a natural way to the divisor such that $D_p \in |L|$; for a general point $p \in L_0$ the divisor $D_p$ is irreducible, and there exist precisely three points (corresponding to the zero values of the $w_i$) for which the divisor is reducible. The map $\psi$ is invariant under the $T^2$-action induced by the moment maps $F_1$ and $F_2$. Moreover, the introduction of $\psi$ allows us to distinguish all orbits of the toric action on $\mathcal{U}$ induced by the pair $F_1$, $F_2$: a pair of values of $F_1$ and $F_2$ (or equivalently, a point in $P_{\mathcal{U}}$) and a point $p \in L_0$ define uniquely an invariant 2-torus in $\mathcal{U}$. Consider the divisor $D_{\rho} \in K_0 \subset |L|$ defined by where $\rho$ is a primitive cube root of unity. This divisor corresponds to the point $p_{\infty}=[1:\rho:\overline \rho] \in L_0$, so we have precisely $\mathcal{U} \setminus D_{\rho}=\psi^{-1}(L_0 \setminus p_{\infty})$.A holomorphic section $\alpha_{\rho}$ with zeros on $D_{\rho}$, given an appropriate Hermitian structure $h$ on $L_D$, defines a smooth function
$$
\begin{equation}
\Psi_{\rho}=-\log\frac{|x_0 y_0+\rho x_1 y_1+\overline \rho x_2 y_2|} {\sqrt{\sum_{i=0}^2|x_i|^2}\sqrt{\sum_{i=0}^2 |y_i|^2}}\,,
\end{equation}
\tag{27}
$$
which is a Kähler potential outside $D_{\rho}$. The following result holds: $\Psi_{\rho}$ is invariant under the Hamiltonian action induced by $F_1$ and $F_2$; the gradient vector field $\lambda_{\rho}$ of the Kähler potential preserves the common level $N(0,0)=\{F_1=F_2=0\}$. In fact, the expression for $\Psi_{\rho}$ is obviously invariant under the Hamiltonian action induced by the $F_i$. Hence, in particular, the critical points of $\Psi_{\rho}$ form invariant subsets of this toric action. On the other hand we have an additional symmetry $\sigma\colon \mathcal{U} \to \mathcal{U}$, which is generated by the transposition $[x_i] \times [y_j] \mapsto [y_i] \times [x_j]$ of direct factors in the product $\mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2 \supset \mathcal{U}$. This transposition clearly preserves the Kähler form, Kähler potential $\Psi_{\rho}$, and the vector field $\lambda_{\rho}$. However, at the same time $\sigma$ interchanges level sets $N(c_1,c_2) \mapsto N(-c_1,-c_2)$, fixing only $N(0,0)$. We return to the Gelfand–Tsetlin Lagrangian spheres $S_i$. Each $S_i$ lies in $N(0,0)$, so we confine ourselves to the map $\psi_0=\psi\big|_{N(0,0)} \to L_0$: it is easy to see that its fibres are invariant 2-tori, with the exception of the three distinguished fibres over
$$
\begin{equation*}
p_1=[1:-1:0],\quad p_2=[0:1:-1],\quad\text{and}\quad p_3=[1:0:-1],
\end{equation*}
\notag
$$
which are circles on which the Hamiltonian vector fields $X_{F_i}$ are collinear. It is easy to see that each $S_i$ is the preimage of the real interval $\gamma_{i+1} \subset L_0$ connecting the two points with indices distinct from $i+1$. Note that just these $S_i$ present non-trivial pairwise different homology classes in $\mathcal{U} \setminus D_{\rho}$. In fact, we can use the first description of $S_0$ as the set of pairs $[x_0: x_1: x_2] \times [-\overline x_0:\overline x_1:\overline x_2]$. Then the intersection $S_0 \cap D_{\rho}$ is defined by the equation which has no non-trivial solutions. Hence $S_0 \cap D_{\rho}=\varnothing$. The complement $\mathcal{U} \setminus D_{\rho}$ if fibred by $\pi_x$, with fibre $\mathbb{C}$ over the first projective plane away from the three points $[1:0:0],\ [0:1:0], and [0:0:1]$ (the projectivizations of eigenspaces of the matrix $A_{\rho}$); the projection $\pi_x(S_0) \subset \mathbb{C}\mathbb{P}^2$, defined by is a sphere separating $[1:0:0]$ from $[0:1:0]$ and $[0:0:1]$, and thus the class $[S_0] \in H_3(\mathcal{U} \setminus D_{\rho}, \mathbb{Z})$ is non-trivial. Repeating these arguments for $S_1$ and $S_2$ we see that the classes $[S_0]$, $[S_1]$, and $[S_2]$ are non-trivial and distinct, so that after selecting suitable orientations they can be represented as $(1,0)$, $(0,1)$, and $(1,1)$ in an appropriate basis of $H_3(\mathcal{U} \setminus D_{\rho},\mathbb{Z})$.The most laborious step is to show that the spheres $S_i$ represent stable classes. To do this we must find finite critical points or subsets for the Kähler potential $\Psi_{\rho}$ and construct the Weinstein skeleton $W(\mathcal{U} \setminus D_{\rho})$. However, as our spheres lie in the level set $N(0,0)$, it is sufficient to reduce the problem to this subset. The Kähler potential $\Psi_{\rho}$ cannot take negative values because the expression under the sign of logarithm is the modulus of the scalar product of two unit vectors. On the other hand it attains the zero value at the point and this is extended to the 2-torus $\psi_0^{-1}([1: \overline \rho:\rho])$ by the $T^2$-action. Now, the set $N(0,0)$ is not smooth: its singular set consists of the three circles $\psi_0^{-1}(p_i)$, and since the gradient vector field $\lambda_{\rho}$ preserves $N(0,0)$, these circles must be invariant, so the field is zero on them. It is easy to see that these circles correspond to saddle points of the general gradient flow, while on $N(0,0)$ they correspond to the maximum value of $\Psi_{\rho}\big|_{N(0,0)}$.Thus, the part of the Weinstein skeleton $W^0(\mathcal{U} \setminus D_{\rho})= N(0,0) \cap W(\mathcal{U} \setminus D_{\rho})$ consists of three components $W_1$, $W_2$, and $W_3$, each of which envelopes the torus $\psi_0^{-1}([1:\overline\rho:\rho])$ and the circle $\psi_0^{-1}(p_i)$ and has the form $\pi^{-1}(S_+)$, where $\pi\colon S^3 \to S^2$ is the Hopf fibration and $S_+ \subset S^2$ is the upper hemisphere. Here $\psi_0^{-1}([1:\overline\rho:\rho])$ corresponds to $\pi^{-1}(\partial S_+)$ and $\psi_0^{-1}(p_i)$ to the preimage of the North pole. From this we obtain closed cycles in $W(\mathcal{U} \setminus D_{\rho})$: the union $W_i \cup W_j$, although it is not smooth, is a closed cycle homeomorphic to $S^3$. It remains to show that for each pair $W_i$, $W_j$ there exists a family of Lagrangian spheres desingularizing the cycle $W_i \cup W_j$. To do this, it is convenient to go over to $L_0$ and describe all the objects under consideration by using $\psi_0$. On $L_0$ we have the following distinguished points:
As shown above, we can obtain Lagrangian spheres not only as the preimages of line segments $[p_i,p_j] \subset L_0$: for each smooth curve $\gamma \subset L_0$ with endpoints $p_i$ and $p_j$ not containing the third point $p_k$ the preimage $\psi_0^{-1}(\gamma)$ is also a Lagrangian sphere. Then, given a union of segments $[p_0,p_i]\cup[p_0,p_j]$, consider a family of smoothing curves $\gamma_t$, $t \in [0,1]$, with fixed endpoints $p_i$ and $p_j$ and such that $\gamma_0=[p_0,p_i] \cup [p_0,p_j]$, $\gamma_1=[p_i,p_j]$, each $\gamma_t$ for $t$ distinct from zero is smooth and does not contain $p_{\infty}$ or $p_k$. Then the family $\psi^{-1}_0(\gamma_t)$ is just a Bohr–Sommerfeld resolution of the closed cycle $W_i \cup W_j$ by a family of Lagrangian spheres (here our problem is considerably simpler because a Lagrangian sphere is always exact). Since in the course of construction we realized all primitive classes in $H_3(\mathcal{U} \setminus D_{\rho},\mathbb{Z})$ by families containing the three spheres $S_0$, $S_1$, and $S_2$, the stable component can contain no other elements. Next, for a reducible divisor $D_1$ represented by the traceless matrix $A_1=\operatorname{diag}(1;1;-2)$ we consider the situation in terms of the map $\psi_0\colon N(0, 0) \to L_0$ again. Now our divisor lies in the fibre over the singular points $p_3=[1:-1:0]$, which simplifies the picture on $L_0$, namely, $p_{\infty}=p_3$, the global minimum is only attained over the point $p_0=\bigl[\frac{1}{2}:\frac{1}{2}:-1\bigr]$ lying on $[p_1,p_2]$, and so the Weinstein skeleton $W(\mathcal{U} \setminus D_1)$ reduces to the Lagrangian sphere $S_2=\psi^{-1}_0([p_1,p_2])$. Since $A_1$ has an eigenvalue of multiplicity two, $H_3(\mathcal{U} \setminus D_1,\mathbb{Z})$ has rank one, and we have thus considered all cases. From these model examples we deduce the same properties of Lagrangian spheres for each irreducible divisor (here we have used the fact that all smooth divisors are symplectically isotopic), and then for each reducible one (here we use its deformation into an irreducible divisor), which gives us the description of the stable component $\mathcal{M}^{\rm st}_{\rm SBS}$ as the set of pairs $(D,\lambda)$, where $\lambda$ is the simple eigenvalue of $A_D$. This yields directly the following algebro-geometric description of $\mathcal{M}^{\rm st} _{\mathrm{SBS}}$. Consider the projective space $\mathbb{C}\mathbb{P}^8$ with homogeneous coordinates $[a_{ij}:z]$ such that $\displaystyle\sum\limits_{i=0}^2 a_{ii}=0$, and let $Y \subset \mathbb{C} \mathbb{P}^8$ be the algebraic subvariety defined by the homogeneous equation $\det(A-z E)=0$ of degree three. Then the projection is well defined: if all the $a_{ij}$ are zero, then $z=0$. Moreover, $\pi^{-1}(D)$ is just the set of eigenvalues of $A_D$. The three-sheeted covering $\pi\colon Y \to \mathbb{C} \mathbb{P}^7$ has the ramification divisor $\Delta \subset Y$, and removing it we obtain $\mathcal{M}^{\rm st}_{\rm SBS}=Y \setminus \Delta$.More precisely, $Y \subset \mathbb{C} \mathbb{P}^8$ is defined by the equation where $P_i$ is a homogeneous polynomial of degree $i$ in the variables $a_{ij}$ (the coefficients of the characteristic equation); and the ramification divisor $\Delta \subset Y$ is given by the additional equation so that $\Delta \in \bigl|\mathcal{O}(2)|_Y\bigr|$.Remarkably, the stable component is again represented by algebro-geometric data. Remark 4. The construction of the stable component acquires remarkable additional meaning for Fano varieties. Since a Fano variety $X$ has by definition the distinguished ample bundle $L=K^*_X$, it is natural to consider for it the corresponding stable component $\mathcal{M}^{\rm st}_{\rm SBS}$ by fixing a suitable topological type of Lagrangian submanifolds. For general divisors $D \in |K_X^*|$ the group $H_n(X \setminus D, \mathbb{Z})$ has the same rank, and fixing a basis $h_1,\dots,h_k$ of it we can define a map of the open part $|K^*_X|^0$ to $\mathbb{C}\mathbb{P}^{k-1}$ by where $\Theta_D$ is the highest-order holomorphic form on the complement $X \setminus D$, which is uniquely defined up to a multiplicative constant, and $\langle\,\cdot\,;\,\cdot\,\rangle$ is the canonical pairing of cohomology and homology classes. This map is particularly interesting in the case of Fano varieties: for a variety of general type the corresponding highest-order form is globally defined, defines a class in $H^n(X,\mathbb{C})$, and considerations of complements provide no additional information. Furthermore, we can link this map with the stable component $\mathcal{M}^{\rm st}_{\rm SBS}$ just for Fano manifolds. For example, for the flag variety $X=F^3$ considered above the bundle $L$ has the following property: $L^2=K^*_{F^3}$, that is, for each divisor $D \in |L|$ we can consider the corresponding highest-order holomorphic form $\theta_D$ with second-order pole on $D$. Then we have the map of the open part where, as above, the $S_i$ are Gelfand–Tsetlin Lagrangian spheres representing primitive non-trivial classes in $H_3(F^3 \setminus D,\mathbb{Z})$ for a general divisor $D$. On the other hand, on the most stable component $ \mathcal{M}_{\rm SBS}^{\rm st}$ we can define a potential function as follows. For the projective space $|K_X^*|$ we consider an arbitrary holomorphic section of the bundle $\mathcal{O}(1)$, which vanishes on the corresponding hyperplane $H \subset |K_X^*|$. Then for each divisor $D \in |K_X^*| \setminus H$ the corresponding highest order holomorphic form $\theta_D \in \Omega^{n,0}(X \setminus D)$ is uniquely determined by the fixed section. Hence we obtain the function for an arbitrary representative of $S$ (as $\theta_D$ is holomorphic, the integral is independent of the choice of $S$).If $K^*_X$ is not just ample, but very ample, then we can introduce an even more interesting object. In this case each point $x \in X$ corresponds to the hyperplane $H_x \subset |K_X^*|$ of divisors containing $x$; the Cartesian product $X \times |K_X^*|$ contains the determinantal divisor $\Delta=\{(x,D) \colon x \in D\}$, and outside it we can fix a section $\widetilde\theta_{\Delta}$ of the corresponding bundle so that for each $x \in X$ and $D \in |K_X^*| \setminus H_x$ we have a well-defined highest order holomorphic form $\theta_D^x$ on $X \setminus D$. Now, removing from $X \times \mathcal{M}^{\rm st}_{\rm SBS}$ the subset
$$
\begin{equation*}
\widetilde\Delta=\{(x,(D,\langle S\rangle)) \colon x \in D\},
\end{equation*}
\notag
$$
we obtain the well-defined function
$$
\begin{equation}
\Theta\colon (X \times \mathcal{M}^{\rm st}_{\rm SBS}) \setminus \widetilde \Delta \to \mathbb{C}, \qquad \Theta(x,D,\langle S \rangle)=\int_S \theta^x_D.
\end{equation}
\tag{31}
$$
The removal of $\widetilde\Delta$ from the product $X \times \mathcal{M}^{\rm st}_{\rm SBS}$ can be realized neatly by considering an analogue of the incidence cycle of the form $\mathcal{D}(X) \subset X \times \mathcal{M}^{\rm st}_{\rm SBS}$, which consists of the elements $(x,D,\langle S \rangle)$ such that the equivalence class $\langle S \rangle$ has a representative $S$ containing the point $x$. In fact, for each representative the intersection $S \cap D$ must be trivial, which shows that $\mathcal{D}(X)$ is just the complement of $\widetilde\Delta$. Thus, in the case of a very ample anticanonical bundle, for a Fano variety $X$ we have a well-defined function $\Theta\colon \mathcal{D}(X) \to \mathbb{C}$ depending on the Hermitian structure on $K^*_X$ and the topological type of Lagrangian submanifolds. A natural question is: what are the properties of this function? 4. Synthesis: variations of parametersThe definition of the stable component $\mathcal{M}_{\rm SBS}^{\rm st}$ is based on SBS-deformations of cycles contained in Weinstein skeletons, but such a deformation is always connected with the corresponding deformation of our main parameter, the prequantization connection $a$, called above the gauge. In fact, if $S_t \subset X \setminus D$ is a smooth Lagrangian BS-submanifold that is close to a closed cycle $S_0 \subset W(X \setminus D)$, then there exists a small variation $a_t=a_I+\delta_t$ of the distinguished gauge such that $S_t$ satisfies the $\alpha$-SBS condition with respect to $a_t$, where the holomorphic section $\alpha \in H^0(X,L)$ is defined by $(\alpha)_0=D$. In fact, since $S_t$ is $D$-exact with respect to the divisor $D$, so that it is exact with respect to the form $\operatorname{Im}\rho(\alpha)$, the restriction of this form is the differential of some function $f \in C^{\infty}(S_t,\mathbb{R})$. As $S_t$ is close to $S_0$, this function must be sufficiently small and can be extended to the whole of $X$ as a smooth function $F \in C^{\infty}(X,\mathbb{R})$. Then the correction term $\delta_t$ is just $-\imath\,\mathrm{d}F$, which, when restricted to $S_t$, compensates for the small additive $\mathrm{d}f$ described above. It is convenient here to give the following definition. Definition 5. Let $S \subset X \setminus D$ be a $D$-exact Lagrangian submanifold with respect to $D$. Given a gauge $a=a_I$, by the distance of $S$ to the Wienstein skeleton $W(X \setminus D)$ we mean the supremum norm of the function $f$ such that $\mathrm{d}f=\operatorname{Im}\rho(\alpha)\big|_S$. It is easy to verify that this is a well-suited term: $S$ lies in $W(X \setminus D)$ if and only if the above norm vanishes; otherwise the norm is positive. In fact, if the supremum norm vanishes, then $f$ is a constant, so that $\operatorname{Im}\rho(\alpha)\big|_S=0$. Reformulating the SBS-condition for the holomorphic case we see that $S$ is included in the Weinstein skeleton. Following [8], we denote this distance by $N_{\lambda}(S)$, where $\lambda$ is the gradient field of the Kähler potential $\Psi_{\alpha}$. It is a remarkable property of $N_{\lambda}(S)$ that this function has no local minima on the space of Hamiltonian deformations of $S$: it only has the global minimum zero. Namely, it was shown in [8] that each exact $S \subset X \setminus D$ has a Hamiltonian deformation $S'$ such that $N_{\lambda}(S)>N_{\lambda}(S')$, provided that the first quantity is distinct from zero. This property suggests a way to proving that, in fact, the stable component $\mathcal{M}^{\rm st}_{\rm SBS}$ coincides with $\widetilde{\mathcal{M}}_{\rm SBS}$. We begin by stating a question. Question 3. Given a topological type of Lagrangian submanifolds, can we find a constant $c>1$ such that an arbitrary exact $S \subset X \setminus D$ has a Hamiltonian deformation $S_1$ such that $N_{\lambda}(S) > c \cdot N_{\lambda} (S_1)$? If such a constant existed, then using Hamiltonian deformations sequentially we could construct a sequence $S_0,S_1,\dots,S_k$ ‘tending’ to the Weinstein skeleton $W(X \setminus D)$ which can be extended to a homotopy connecting $S_0$ with a cycle in the skeleton. In this way, for an arbitrary exact $S_0$ we would establish the stability of the corresponding class $\langle S_0 \rangle$. In addition, an affirmative answer to the last question would imply both conjectures of Eliashberg — see [8]. The definition of $N_{\lambda}(S)$ agrees well with the natural distance function on the gauge space $\mathcal{O}_{\omega}$. For a pair of gauges $a, a' \in \mathcal{O}_{\omega}$ the difference between the connections $a$ and $a'$ is a global exact 1-form of the form $\imath\,\mathrm{d}F$, where $F \in C^{\infty}(X,\mathbb{R})$, and it is natural to set the distance $d(a,a')$ to be the supremum norm of $F$, the difference between the maximum and minimum of $F$ on the whole of $X$. It is easy to see that the gauge $a_t$ such that $S_t$ is $\alpha$-SBS with respect to it must satisfy the inequality Moreover, it can always be chosen so that
$$
\begin{equation*}
\max_{x \in X} F_t=\max_{x \in S_t} f\quad\text{and} \quad \min_{x \in X} F_t=\min_{x \in S_t} f,
\end{equation*}
\notag
$$
and, in addition, $\mathrm{d}F=0$ outside a small neighbourhood of $S_t$, so that inequality (32) turns to equality. We have the following result for this situation. Proposition 8. A closed cycle $S_0 \subset W(X \setminus D)$ admits a BS-resolution by a homotopy $\{S_t\}$ if and only if the gauge $a_I$ has a deformation $a_t$, $t \in [0,1]$, $a_0=a_I$, such that for each $t > 0$ the preimage $q_1^{-1}([\alpha]) \subset \mathcal{U}_{\rm SBS}(a_t)$ is non-empty. Now, it follows from the properties of $q_1$ that, in fact, the preimage is non-empty not only for some family of gauges $a_t$, but for a sufficiently general variation of the distinguished gauge $a_I$. Precisely this property can be used for the synthesis of constructions presented above and for a consistent definition of a finite-dimensional moduli space of special BS-Lagrangian submanifolds, which is our main task. Thus, the synthetic construction proposed in [8] looks as follows. Consider a simply connected compact smooth (projective) algebraic variety $ X$ with a very ample linear bundle $L \to X$, which must exist by definition. Choosing an appropriate Hermitian structure $h$ on $L$ we consider the corresponding Hermitian form $\omega_h$ defined by a system of Kähler potentials $\psi_{\alpha}=-\log |\alpha|_h$ produced by holomorphic sections $\alpha \in H^0(X, L)$ on the complements $X \setminus D_{\alpha}$. On the other hand our choice of $h$ in the presence of a holomorphic structure on $L$ distinguishes a connection $a_I \in \mathcal{O}(\omega_h)$. Thus, our choice of an appropriate $h$ takes us to the situation considered above in § 1: we have the prequantization quadruple $(M=X,\omega=\omega_h,I,L,a_I)$ and can use the constructions of SBS-geometry. In addition, from § 2 we know details of these constructions in the holomorphic situation, when for each class of holomorphic sections $[\alpha] \in \mathbb{P} H^0(X,L) \subset \mathbb{P} \Gamma(X,L)$ the preimage $q_1^{-1}([\alpha]) \in \mathcal{U}_{\rm SBS}(a_I)$ can explicitly be described as follows. Considering the corresponding Kähler potential $\psi_{\alpha}$ on the complement $X \setminus D_{\alpha}$, we look at its critical points $x_1,\dots,x_N$ and select only the finite trajectories of the gradient flow induced by the gradient vector field $\operatorname{grad}\psi_{\alpha}$, which connect points $x_i$ pairwise (while infinite trajectories approach the pole on $D_{\alpha}$); the union of all these finite trajectories forms the Weinstein skeleton of the complement $W(X \setminus D_{\alpha})$, and then the smooth BS-submanifold $S \subset X \setminus D_{\alpha}$ is special with respect to the class $[\alpha]$ if and only if $S$ lies in $W(X \setminus D_{\alpha})$. On the other hand, as we know from [12], even in simplest cases $W(X \setminus D_{\alpha})$ has no smooth components of highest dimension, so the straightforward preimage $q^{-1}_1([\alpha])$ is just empty. At the same time this holomorphic situation also shows that if the Weinstein skeleton has dimension $n$, then it contains non-trivial $n$-cycles, which form the homology base for the special BS-submanifolds in the case described above in § 3. If we distinguish a primitive class in $W(M \setminus D_{\alpha})$ for a general holomorphic section, then such a class exists in the corresponding Weinstein skeletons also for other sufficiently general sections, and if such a cycle has an SBS-resolution for some general holomorphic section, then it also exists for other general sections. We return to our previous observations. The preimage is empty because the situation treated in the thesis (see (15)) is too far from a general one: it is very special because the choice of a prequantization connection $a_I$ and that of a holomorphic section $\alpha$ are too intimately connected. If we allow a smooth perturbation of the gauge $a_I \to a_t$, then smooth SBS-manifolds can appear. For more convenient statements we use deformations of holomorphic sections in place of deformations of the connection $a_I$: as shown in § 1 above, a deformation $a_I \mapsto a_I+\imath\,\mathrm{d}F$ is equivalent to the deformation $\alpha \to e^{\imath F} \alpha$ of the section, which preserves its set of zeros. Moreover, in place of the ‘one-dimensional’ deformation of the connection we can consider a ‘multidimensional’ deformation of holomorphic sections, by introducing the correspondence
$$
\begin{equation}
\delta\colon \mathbb{P} H^0(X,L) \to C^{\infty}(X,\mathbb{R}_+), \qquad [\alpha] \mapsto [\delta([\alpha])\alpha] \in \mathbb{P}\Gamma(X,L),
\end{equation}
\tag{33}
$$
which preserves the zeros of sections but transforms holomorphic sections into more general ones, for which it would be natural to believe that the corrected preimage is non-trivial. In the case under consideration, by a ‘small’ perturbation we mean the following: as each $\delta([\alpha])$ is a global smooth function on the compact manifold $X$, we can use the universal bound $\max\delta(p)-\min\delta(p) \leqslant \varepsilon$ for some small $\varepsilon$ for each $p$. Moreover, we consider only deformations distinct from a constant in a sufficiently small neighbourhood of the skeleton $W(X \setminus D_{\alpha})$. Let $\mathbb{P}H^0(X,L)_{\delta} \subset \mathbb{P}\Gamma (X,L)$ denote the corresponding deformation of the original finite-dimensional projective space. It is easy to see that it has the following property. Proposition 9. For a sufficiently small deformation the space $\mathbb{P} H^0(X,L)_{\delta}$ is a smooth real submanifold of dimension $2(h^0(X,L)-1)$, which is symplectic with respect to the Kähler form $\Omega_{\rm FS}$. In fact, the projective space $\mathbb{P}H^0(X,L)$ is associated with the family $\{\mathbb{P}^0(D_{\alpha})\}$ of pairwise disjoint affine subspace of $\mathbb{P}\Gamma(X,L)$, each of which consists of classes of sections with fixed sets of zeros. It is easy to see that all these affine subspaces are transversal to $\mathbb{P}H^0(X,L)$. Transformations induced by $\delta$ act along the ‘slices’ $\mathbb{P}^0(D_{\alpha})$, which ensures smoothness, and as the deformations are sufficiently small, the resulting submanifold $\mathbb{P}H_0(X,L)_{\delta}$ is still symplectic (for details, see [8]). We return to the pair $(X,L)$ described above. For a suitable Hermitian structure $h$ we propose the following definition. Definition 6. The moduli space of SBS-cycles is equal to where $\delta$ is a general sufficiently small deformation, $\operatorname{top}S$ is the topological type of $S$, and $[S] \in H_n(X,\mathbb{Z})$ is a fixed homology class. We must show that this definition is consistent, that is, the geometry of the moduli space (which we will denote by $\mathcal{M}_{\rm SBS}$ for short, provided that the topological type and homology class are clear from the context) is independent of the choice of a small deformation. We have the following result. Proposition 10. The space $\mathcal{M}_{\rm SBS}$ is independent of the choice of a general deformation. The proof follows from the comparison of $\mathcal{M}_{\rm SBS}$ with a universal object independent of any deformations, the stable component of the moduli space of $D$-exact Lagrangian submanifolds, which we introduced in the preceding section (see Definition 4). Namely, the following result was proved in [8]. Theorem. The moduli space $\mathcal{M}_{\rm SBS}$ defined above is naturally isomorphic to the stable component $\mathcal{M}^{\rm st}_{\rm SBS}$ of the moduli space of $D$-exact Lagrangian submanifolds. We sketched the idea of the proof in the beginning of this section. The statement itself has a simple geometric interpretation: a Bohr–Sommerfeld resolution of the $n$-cycle $S_0 \subset W(X \setminus D_{\alpha})$ can be associated either with $\delta$-deformations of the first components or with Hamiltonian isotopies of the second components of the pairs $(p_t,S_t) \in \mathcal{U}_{\rm SBS}$. Schematically, the correspondences look as follows: Here we denote by upper commas the virtual moduli space, which can turn out to be empty, and the special arrow $\rightsquigarrow$ denotes a deformation.Thus the definition of the moduli space of SBS-Lagrangian submanifolds is consistent. However, the following question arises here in a natural way: why should we at all return to special Bohr–Sommerfeld geometry if the problem of the construction of finite-dimensional moduli space that we stated in the introduction has already found a decent solution in the previous section, in the form of the stable component $\mathcal{M}^{\rm st}_{\rm SBS}$? Moreover, the simple description of a Kähler structure of this variety also ensures the important property we hoped for at the beginning of our journey. However, examples presented above show that the moduli space can also admit the special structure of an algebraic variety. This is why our main conjecture is that $\mathcal{M}_{\rm SBS}=\mathcal{M}^{\rm st} _{\mathrm{SBS}}$ can be represented in the form $Y \setminus D$, where $Y$ is a projective algebraic variety and $D$ is an ample divisor. What is actually the difference between the thesis and antithesis stated above? We return to the very beginning and the definition of $\mathcal{U}_{\rm SBS}(a)$ as a subspace of the Cartesian product $\mathbb{P}\Gamma(M,L) \times \mathcal{B}_S$ in the most general case of a symplectic manifold $M$ with integer form $\omega$. On each direct factor we have a natural $\mathrm{U}(1)$-fibration. For the first factor it is $\mathcal{O}(1)$, a standard linear bundle on a projective space. Since the Kähler structure on this space is fixed (recall that we have an Hermitian scalar product on the original vector space $\Gamma(M,L)$), this bundle is endowed with a distinguished Hermitian connection $A$ with curvature form $F_A=2\pi\imath\Omega_{\rm FS}$, where $\Omega_{\rm FS}$ is the corresponding Kähler form of the Fubini–Study metric. Note that the original $\mathrm{U}(1)$-action on the prequantization bundle $L \to M$ induces also the corresponding action on $\mathcal{O}(1)$. On the other hand the moduli space $\mathcal{B}_S$ is the base of the natural $\mathrm{U}(1)$-bundle $\mathcal{P}_S(a) \to \mathcal{B}_S$ with total space consisting of Planck cycles in the contact manifold $\operatorname{tot}(S^1(L) \to M)$ (see [2]): the fibre over $S \in \mathcal{B}_S$ consists of covariantly constant lifts of $S$ to $(S^1(L),a)\big|_S$, and the corresponding $\mathrm{U}(1)$-action is induced by the same basic $\mathrm{U}(1)$-action on $L \to M$. Note that the bundle $\mathcal{P}_S(a) \to \mathcal{B}_S$ depends on the prequantization connection $a \in \mathcal{O}(\omega)$, because each fibre depends on this connection. Thus we have two $\mathrm{U}(1)$-bundles, $q_1^* \mathcal{O}(1)$ and $q_2^* \mathcal{P}_S(a)$, on $\mathbb{P} \Gamma (M, L) \times \mathcal{B}_S$. Then the subset $\mathcal{U}_{\rm SBS}(a)$ of the Cartesian product can be described as follows. Proposition 11. The bundles $q_1^* \mathcal{O}(1)$ and $q_2^* \mathcal{P}_S(a)$ become canonically isomorphic after the restriction to $\mathcal{U}_{\rm SBS}(a)$. In fact, the fibre of the first bundle over a point $(p,S)$ consists of the elements $e^{\imath t} \alpha^*$, which are pointwise dual on $S \subset M$ to a section $\alpha$ such that $p=[\alpha]$. On the other hand the fibre of the second bundle consists of the elements $e^{\imath t}\sigma_S$. Hence, if in the product of fibres $\mathrm{U}(1)\times \mathrm{U}(1)$ we can distinguish canonically a diagonal group $\mathrm{U}(1)$ by means of a condition not involving any additional choices, then we obtain a canonical identification. This natural condition reads: the natural pairing $\alpha^*\big|_S(\sigma_S) \in C^{\infty}(S,\mathbb{R}_+)$ is a real positive function on the whole of $S$; however, this is precisely the condition defining a special submanifold and holding at all points in the cycle $\mathcal{U}_{\rm SBS}(a)$ by its definition. Moreover, it is easy to see that $\mathcal{U}_{\rm SBS}(a)$ is the maximal possible set over which the bundles pulled back are canonically isomorphic. Hence $\mathcal{U}_{\rm SBS}(a)$ is the base of a natural $\mathrm{U}(1)$-bundle $\mathcal{L}$ that is canonically isomorphic to the restrictions of the pullbacks $q_1^*\mathcal{O}(1)$ and $q_2^*\mathcal{P}_S(a)$. In particular, $\mathcal{L} \to \mathcal{U}_{\rm SBS}(a)$ carries a Hermitian connection $\widetilde A=q_1^* A$ with curvature form and this form is Kähler in the weak sense. Hence any other Hermitian connection on $\mathcal{L}$ is defined by an appropriate 1-form on $\mathcal{U}_{\rm SBS}(a)$, and it is a natural problem to find a suitable correction for $\widetilde A$ pulled back from $\mathcal{B}_S$ to make the resulting connection have a curvature proportional to a Kähler form in the strong case or to a symplectic form.Now, by definition (see Definition 5 above) the moduli space $\mathcal{M}_{\rm SBS}$ is embedded in $\mathcal{U}_{\rm SBS}(a_I)$, so the natural restriction $\mathcal{L}_{\delta} \to \mathcal{M}_{\rm SBS}$ is defined up to a choice of a suitable small deformation $\delta$. However, it is easy to see that, topologically, the restriction of $\mathcal{L}_{\delta}$ is independent of the choice of $\delta$, so, rather than just some variety $\mathcal{M}_{\rm SBS}$ (as in the case of $\mathcal{M}^{\rm st}_{\rm SBS}$), we have a pair of the form For this bundle the variations $\delta$ correspond to variations of the Hermitian structure because they can be expressed in terms of variations of the basic connection $a_I$, which, as we have seen, correspond to variations of covariantly constant sections $\sigma_S$. These latter correspond in their turn to variations of the local basis in a neighbourhood of the point in question, which in the end corresponds to variations of the Hermitian structure.On the other hand our conjecture that the moduli space is algebraic was originally put forward in [10] with regard to the stable component $\mathcal{M}^{\rm st}_{\mathrm{SBS}}$ of the moduli space of exact Lagrangian submanifolds. However, the difference between algebraic and Kähler cases lies in the existence of an ample linear bundle, so for the treatment of this conjecture it is natural to use the bundle $\mathcal{L} \to \mathcal{M}_{\rm SBS}$: if, fixing an appropriate Kähler structure on the base, we find a Hermitian connection on $\mathcal{L}$ with curvarure form proportional to the Kähler form and also find a section without zeros on $\mathcal{M}_{\rm SBS}$, then this will produce the required algebraic structure. To implement this program we must develop many technical details, starting with questions on connections on $\mathcal{L}$, outlined already in [8]. Even if the conjecture is not confirmed, the techniques developed on this way can be useful in the future, in studies of the differential geometry of the moduly variety $\mathcal{B}_S$, with a view to apply them to geometric quantization constructions. Note that if our conjecture that $\mathcal{M}_{\rm SBS}$ is algebraic is confirmed, this will have interesting consequences even in the context of algebraic geometry. Namely, consider a projective variety $X_0$ and an ample divisor $D_0 \subset X_0$. As we have seen, such a pair induces a moduli space $\mathcal{M}_{\rm SBS}$; by the conjecture of algebraicity this variety has the form $X_1 \setminus D_1$, where $(X_1,D_1)$ is a pair of just the same form as the original one. Hence from $(X_1,D_1)$ we can construct another $X_2$ with ample divisor $D_2$, and so on. As a result, we obtain a chain of projective varieties defined in terms of the Lagrangian geometry of these varieties. We conclude by another brief observation. Returning to the beginning of this survey we recall one of the central ideas of [4], namely, the correspondence
$$
\begin{equation*}
\text{holomorphic bundles}\ \ \longleftrightarrow\ \ \text{Lagrangian submanifolds}
\end{equation*}
\notag
$$
between very different geometric objects, which seeks to actualize in geometric constructions. Given a variety $X$, assume that we have constructed the moduli space $\mathcal{M}_{\rm SBS}$. Then any Lagrangian submanifold $K \subset X$ induces a bundle $E_K \to \mathcal{M}_{\rm SBS}$: since each element of the moduli space is represented by a class of $D$-exact Lagrangian submanifolds $\langle S \rangle$, as a fibre we take the space $CH(K,S,\mathbb{C})$, where $S$ is a representative of $\langle S \rangle$, and $CH$ denotes the Floer cohomology of the pair $(K,S)$. Since stability under Hamiltonian deformations is the fundamental property of this cohomology, fibres of $E_K$ are well defined. Thus, our constructions follow the directions outlined in [4]. 
Citation in format AMSBIB
\Bibitem{Tyu25} |
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