N. A. Tyurin, “Special Bohr–Sommerfeld geometry: variations”, Izv. RAN. Ser. Mat., 87:3 (2023), 184–205; Izv. Math., 87:3 (2023), 595

We first recall basic constructions of special Bohr–Sommerfeld geometry of compact symplectic manifolds (details can be found in [1]–[4]).

Let $(M, \omega)$ be a compact simply connected symplectic manifold of dimension $2n$ such that the symplectic form $\omega$ is integer, and so its cohomology class is integer. Consider the corresponding complex line bundle $L \to M$ such that $c_1(L)=[\omega]$ equipped with a Hermitian structure $h$, therefore, the corresponding Hermitian connection space $\mathcal{A}_h(L)$ contains the subset $\mathcal{O}(\omega)$ consisting of the solutions of the equation $F_a=2 \pi i \omega$ (in the simply connected space, this set $\mathcal{O}(\omega)$ is just an orbit under the gauge group action). We choose an element $a \in \mathcal{O}(\omega)$ and combine the corresponding prequantization pair $(L, a)$, which plays an essential role in Geometric quantization (GQ); from the GQ-constructions, we take the Hilbert space $\Gamma (M, L)$ consisting of smooth sections of $L$, where the Hermitian scalar product $\langle s_1, s_2\rangle =\int_M (s_1, s_2)_h \,d \mu_L$ is defined by $h$ and the Liouville form $d \mu_L$.

We fix a topological type $\operatorname{top} S$ of a smooth orientable $n$-dimensional manifold and the class of middle homology $[S] \in H_n(M, \mathbb{Z})$. Hence we get the corresponding moduli space $\mathcal{B}_S$ of Bohr–Sommerfeld (BS, for short) Lagrangian cycles of fixed type (see [5]), which is an infinite dimensional Fréchet-smooth real manifold, where points can be understood as Lagrangian submanifolds $S \subset M$ of the above fixed topological type that satisfy the Bohr–Sommerfeld condition: for each $a \in \mathcal{O}(\omega)$, the pair $(L,a)$, when restricted to $S$, admits covariantly constant sections. The moduli space $\mathcal{B}_S$ is also used in a different approach to Geometric quantization (known as the Lagrangian approach, see [6]). Here and in what follows, we consider only the case of smooth Lagrangian submanifolds.

Leaving aside of our discussion the motivation and communicative constructions of Geometric quantization, we define a certain universal object $\mathcal{U}_{\mathrm{SBS}}(a)$ in the direct product $\mathbb{P} \Gamma (M, L) \times \mathcal{B}_S$ by the following rule. A pair $(p=[\alpha], S)$, where $\alpha \in \Gamma(M, L)$ is a smooth section, which represents the equivalence class $p$, belongs to $\mathcal{U}_{\mathrm{SBS}}(a)$ if the proportionality coefficient $\alpha|_S/\sigma_S$ has the form $e^{\imath c} f$, here $\sigma_S$ is a covariantly constant section of $(L,a)|_S$, $c$ is a real constant, and $f \in C^{\infty}(S, \mathbb{R}_+)$ is a strictly positive real function on $S$. Note that a change in $a \in \mathcal{O}(\omega)$ is reflected in the definition, since the covariantly constant section $\sigma_S$ does depend on the choice of $a$ for a fixed Bohr–Sommerfeld submanifold $S$.

By the very definition, $\mathcal{U}_{\mathrm{SBS}}(a)$ admits two natural projections:

The second projection $p_2\colon \mathcal{U}_{\mathrm{SBS}}(a) \to \mathcal{B}_S$ splits into the composition $\pi \circ \tau$, where $\pi\colon T \mathcal{B}_S \to \mathcal{B}_S$ is the canonical projection of the tangent bundle, and the resting map $\tau\colon \mathcal{U}_{\mathrm{SBS}}(a) \to T \mathcal{B}_S$ has Kahler fibers (details can be found in [4]).

Now if we turn to the natural problem, which has appeared many times in both symplectic geometry and mathematical physics, and is devoted to possible constructions of finite dimensional moduli spaces of Lagrangian submanifolds satisfying certain special properties, then the special Bohr–Sommerfeld geometry (SBS, for short) can be exploited in this way. Namely suppose that, from some natural conditions, a finite dimensional subspace $\mathbb{P}^N$ is singled out in $\mathbb{P} \Gamma (M, L)$. Then its preimage with respect to the first projection $p_1^{-1}(\mathbb{P}^N)$ should be a finite dimensional Kahler manifold. Moreover, an appropriate $\mathbb{P}^N \subset \mathbb{P} \Gamma (M, L)$ can be chosen almost automatically in the case when $(M, \omega)$ is equipped with a compatible integrable complex structure $I$. Indeed, the compactness of $M$ implies that the space of holomorphic sections $H^0(M_I, L)$ of the prequatization bundle is finite dimensional; therefore, one could take $\mathbb{P}^N=\mathbb{P} H^0(M_I, L) \subset \mathbb{P} \Gamma (M, L)$, and define naturally the preimage $p_1^{-1}(\mathbb{P} H^0(M_I, L))$ as a distinguished geometrical object, which characterizes the Kahler nature of $(M, \omega, I)$.

This construction turns out to be especially natural in the case of algebraic varieties. Indeed, a compact algebraic variety $X$ with very ample line bundle $L$ (note that such a bundle $L$ must exist by definition) is presented as $(M, \omega, I)$ from the real geometry point of view, where $c_1(L)=[\omega]$; this correspondence can be realized by the choice of an appropriate Hermitian structure $h$ on $L$ such that $\omega=-d I\, d (\ln | \alpha |_h)$ on the complement of the zero set $D_{\alpha}=\{ \alpha=0 \} \subset X$ for a holomorphic section $\alpha \in H^0(X, L)$. It is clear that such $\omega$ is not unique, but the induced Lagrangian geometries for different forms of this type are equivalent (at the same time, for different principal polarizations $L_1$ and $L_2$ the corresponding geometries can be different). In the presence of the complex structure $I$ one has a distinguished Hermitian connection $a_I \in \mathcal{O}(\omega)$ compatible with the holomorphic structure on $L$, and the SBS-construction leads to the first coarse definition: the moduli space $\mathcal{M}_{\mathrm{SBS}}(c_1(L))$ of special Bohr–Sommerfeld Lagrangian submanifolds is $p^{-1}(\mathbb{P} H^0(M_I, L)) \subset \mathcal{U}_{\mathrm{SBS}}(a_I)$.

According to the general theory, this coarse moduli space must be a finite dimensional smooth Kahler manifold, however such a direct and coarse approach leads to a trivial answer due to the following reasons. In [2], it was shown that a Bohr–Sommerfeld Lagrangian submanifold $S$ is special with respect to holomorphic section $\alpha \in H^0(M_I, L)$ under the choice, as the prequantization connection, of distinguished connection $a_I$ in the definition of $\mathcal{U}_{\mathrm{SBS}}$ if and only if this submanifold is contained in the Weinstein skeleton of the complement $M_I \setminus D_{\alpha}$ formed by finite trajectories of the gradient flow of the function $-\ln | \alpha |_h$, while the zero divisor $D_{\alpha}$ of the section $\alpha$ attracts all semi-infinite trajectories, see [7]. And even in the most simple cases, the Weinstein skeletons for holomorphic sections do not admit smooth Lagrangian components (in [3], such an example involving $M_I=\mathbb{C} \mathbb{P}^1$ and $L=\mathcal{O}(3)$ was presented — even in this simplest case, smooth loops contained by the corresponding to generic holomorphic section Weinstein skeletons do not exist!). Therefore, the above coarse definition leads to trivial answers.

At the same time, parallel results were obtained in [3]: in the same situation, we constructed moduli spaces of $D$-exact Lagrangian submanifolds which are presented by finite dimensional Kahler manifolds fibered over open parts of $\mathbb{P} H^0(M_I, L)$ with discrete fibers. These moduli spaces were denoted by $\widetilde{\mathcal{M}}_{\mathrm{SBS}} (c_1(L))$ in [3], since in the same paper we claimed that these moduli spaces are related to SBS-geometry.

The main aim of the present paper is to provide a correction of the coarse definition of the moduli space $\mathcal{M}_{\mathrm{SBS}}(c_1(L))$. It turns out that such a correction is quite natural since there exists a broad space of possible deformations depending on the choice of a Hermitian prequantization connection. At the same time, in this way, we find an explicit correspondence between the “special Bohr–Sommerfeld cycles” and the “equivalence classes of Hamilton isotopic $D$-exact Lagrangian submanifolds”, and show how this correspondence helps in the work on the our main conjecture states that the moduli space of special Bohr–Sommerfeld Lagrangian cycles for an algebraic variety is not only Kahler, but is algebraic itself.

At the same time, the resulting relationship between SBS-geometry and the theory of Liouville vector fields and Weinstein structures suggests new approaches to some problems in the theory, and, in particular, to the Eliashberg conjectures on smooth exact Lagrangian submanifolds. In the case, where an open symplectic manifold is presented as the complement $M \setminus D_{\alpha}$ of the zero set of a certain regular section of the prequantization bundle and the Liouville vector field $\lambda_{\alpha}$ can be completed to Weinstein structure $(\phi, \lambda_{\alpha})$, the compact exact Lagrangian submanifold $S \subset M \setminus D_{\alpha}$ must be Bohr–Sommerfeld. This means that the above moduli space $\mathcal{B}_S$ splits naturally into connected components $\mathcal{B}_S^i$, whose elements are Lagrangian submanifolds are disjoint from $D_{\alpha}$. If an element $S$ is exact, then the entire component $S \subset \mathcal{B}_S^i$ consists of exact Lagrangian submanifolds, and so one can naturally define a certain smooth real distance function $N\colon \mathcal{B}_S^i \to \mathbb{R}_{\geqslant 0}$ to the Weinstein skeleton $W(M \setminus D_{\alpha})$. The value of the function $N(S)$ is the difference $\max_{p \in S} f-\min_{p \in S} f$, where $f \in C^{\infty}(S, \mathbb{R})$ is defined by the condition $d f=\omega^{-1}(\lambda_{\alpha})|_S$ for exact $S$. The zero value $N(S)=0$ is equivalent to $S \subset W(M \setminus D_{\alpha})$ (and so to the regularity of $S$). Below, in § 6 we will show that the function $N(S)$ does not admit local minima except the global one with zero value, which implies that, for each $S$ not contained in $W(M \setminus D_{\alpha})$, there always exists a Hamiltonian deformation $S'= \phi^t_H(S)$ such that $N(S')<N(S)$; if in the future we will be able to find an effective bound on a possible decrease of the value of $N(S)$, this should imply existence of a Hamiltonian isotopy $S_t$, $t \in [0, \infty)$, such that $S_0=S$, and, say, $N(S_t)=t^{-1}$, therefore, in the limit we should get a homologically non-trivial cycle in $W(M \setminus D_{\alpha})$, which should lead to a positive answer to the Eliahsberg conjecture on homological non-triviality of smooth exact Lagrangian submanifolds.

§ 1. Geometric interpretation

Consider a compact simply connected symplectic manifold $(M, \omega)$ of real dimension $2n$. Let the cohomolgy class of the symplectic form $[\omega]$ be integer valued. Then there exists a complex line bundle $L \to M$ with the first Chern class $c_1(L)=[\omega]$ (called a prequantization bundle). So, by choosing a Hermitian structure on it we get a Hermitian connection space $\mathcal{A}_h(L)$ together with a distinguished subset $\mathcal{O}(\omega)$ consisting of $a$’s such that the curvature form $F_a=2 \pi \omega$. Since $M$ is simply connected, the subset $\mathcal{O}(\omega)$ is an orbit under the gauge group action. For any Lagrangian submanifold $S \subset M$ and any connection $a \in \mathcal{O}(\omega)$ the restriction $(L, a)|_S$ is a flat line bundle, and we say that $S$ is Bohr–Sommerfeld (BS, for short) if $(L, a)|_S$ admits a covariantly constant section $\sigma_S$, defined uniquely up to scaling by constants. BS-property of $S$ does not depend on the choice of a particular connection $a \in \mathcal{O}(\omega)$ (while $\sigma_S$ does depend on it). Indeed, for any other connection $a_1 \in \mathcal{O}(\omega)$, the difference between the induced covariant derivatives is $\nabla_{a_1}-\nabla_a=\imath\, d \phi$, where $\phi \in C^{\infty}(M, \mathbb{R})$, and so the new covariantly constant section $\sigma^1_S$ must have the form $e^{-\imath \phi} \sigma_S$.

Therefore, the definition of the subset $\mathcal{U}_{\mathrm{SBS}} \subset \mathbb{P} \Gamma (M, L) \times \mathcal{B}_S$, where $\mathcal{B}_S$ is the moduli space of Bohr–Sommerfeld Lagrangian submanifolds, does depend on the choice of a prequantization connection $a \in \mathcal{O}(\omega) \subset \mathcal{A}_h (L)$, since our specialty condition

For a pair of different connections $a_2, a_1 \in \mathcal{O}(\omega)$, the corresponding $\mathcal{U}_{\mathrm{SBS}} (a_i)$ have the following intersection:

Recall that any smooth function $F$ on $M$ induces a certain vector field $\Theta(F)$ on the moduli space $\mathcal{B}_S$: at a point presented by $S$, its value is given by the exact form $d (F|_S)$, and the corresponding flow generated by $\Theta(F)$ is exactly the realization of the motion of $S$ in $M$ under the action of the flow generated by the Hamiltonian vector field $X_F$ (details can be found in [5], [6]). Therefore, intersection (1) can be described geometrically in the following way: the function of the connection difference $\phi$ induces the corresponding vector field $\Theta(\phi) \in \operatorname{Vect} \mathcal{B}_S$, and if one considers the second canonical projection $p_2\colon \mathcal{U}_{\mathrm{SBS}}(a_1) \to \mathcal{B}_S$, then the intersection (1) is the preimage $p_2^{-1}((\Theta(\phi))_0)$ of the singular subset of the vector field $\Theta(\phi)$.

This implies that the intersection condition is essentially imposed on the second element of the pair $(p, S)$ only, which suggests some geometrical interpretation of the subset $\mathcal{U}_{\mathrm{SBS}}(a)$ itself.

Both direct summands $\mathbb{P} \Gamma(M, L)$ and $\mathcal{B}_S$ underly natural $U(1)$-bundles. For the first direct summand, this bundle is $\mathcal{O}(1)$, that is, the standard line bundle over the projective space. Since the Kahler structure on our projective space is fixed (recall that we do have a Hermitian scalar product on the original vector space $\Gamma (M, L)$), this bundle is equipped with a distinguished Hermitian connection $A$ with curvature form $F_A=2 \pi i \Omega_{\mathrm{FS}}$, where $\Omega_{\mathrm{FS}}$ is the corresponding Kahler form of the Fubini–Study metric. Note that the original $U(1)$-action on the prequantization bundle $L \to M$ induces the corresponding action on $\mathcal{O}(1)$.

On the other hand, the moduli space $\mathcal{B}_S$ a underlies certain natural $U(1)$-bundle $\mathcal{P}_S(a) \to \mathcal{B}_S$ formed by Planckian cycles in the contact manifold $\operatorname{tot} (S^1(L) \to M)$ (see [5]): the fiber over a point $S \in \mathcal{B}_S$ consists of covariantly constant lifts of $S$ to $(S^1(L),a)|_S$, and the corresponding $U(1)$-action is induced again by the same original $U(1)$-action on $L \to M$; note that the bundle $\mathcal{P}_S(a) \to \mathcal{B}_S$ depends on the choice of the prequantization connection $a \in \mathcal{O}(\omega)$ since its fibers depend on this choice.

Thus, on the direct product $\mathbb{P} \Gamma (M, L) \times \mathcal{B}_S$ we have two $U(1)$-bundles $p_1^* \mathcal{O}(1)$ and $p_2^* \mathcal{P}_S(a)$. Hence $\mathcal{U}_{\mathrm{SBS}}(a)$ in this direct product can be described as follow.

Indeed, the fiber of the first one over a point $(p, S)$ is presented by elements of the form $e^{\imath t} \alpha^*$, which are over $S \subset M$ pointwise dual to a section $\alpha$ such that $p= [\alpha]$. At the same time, the fiber of the second bundle is presented by elements of the form $e^{\imath t} \sigma_S$; thus, if in the direct product $U(1) \times U(1)$ of these fibers one can canonically distinguish a certain diagonal $U(1)$ which does not depend on any other choices, then we get the canonical identification. This canonical property is the following: the natural pairing $\alpha^*|_S(\sigma_S)\,{\in}\, C^{\infty}(S, \mathbb{R}_+)$ gives a real strictly positive function along the entire $S$, but it is precisely our specialty condition, which is satisfied along $\mathcal{U}_{\mathrm{SBS}}(a)$ by definition.

Moreover, it is not hard to see that the subset $\mathcal{U}_{\mathrm{SBS}}(a)$ is a maximal possible one over which our lifted bundles are canonically isomorphic.

In particular, the bundle $\mathcal{L} \to \mathcal{U}_{\mathrm{SBS}}(a)$ carries the Hermitian connection $\widetilde A=p_1^* A$ with the curvature form $F_{\widetilde A}=2 \pi i p_1^* \Omega_{\mathrm{FS}}$, where the last form is Kahler in a weak sense. Consequently, any other Hermitian connection on $\mathcal{L}$ is defined by the corresponding 1-form on $\mathcal{U}_{\mathrm{SBS}}(a)$, and a natural problem is to find an appropriate corrections to $\widetilde A$ lifted from $\mathcal{B}_S$ such that the corrected connection would have the curvature form proportional to a Kahler one in the strong sense or to a symplectic form.

§ 2. Transformations

On the other hand, it is not hard to see that all $\mathcal{U}_{\mathrm{SBS}}(a)$ are isomorphic to each other. Indeed, for any real smooth function $F \in C^{\infty}(M, \mathbb{R})$, which is globally defined over the whole $M$, there exists a corresponding projective transformation $P(F) \in \operatorname{Aut} \mathbb{P} \Gamma (M, L)$ given by $[\alpha] \mapsto [e^{\imath F} \alpha]$. Since on the level of the vector space $\Gamma(M, L)$ this transformation preserves our Hermitian scalar product

For a pair of Hermitian prequantization connections $a_1$, $a_2$ such that the difference is $\nabla_{a_2}-\nabla_{a_1}=\imath\, d F$ with a smooth function $F \in C^{\infty}(M, \mathbb{R})$, we have the following equivalence: $([\alpha], S) \in \mathcal{U}_{\mathrm{SBS}}(a_1)$ if and only if $(P(F)([\alpha]), S) \in \mathcal{U}_{\mathrm{SBS}}(a_2)$. Indeed, $\alpha|_S=e^{\imath c} f \sigma_S^1=e^{\imath c} f e^{-\imath F|_S} \sigma_S^2$ if and only if $e^{\imath F} \alpha|_S=e^{\imath c} f \sigma_S^2$. We this have the following result.

Here, we essentially use the projective nature of the first component of the subset element. More difficult is the question of deformations of the second component (which is a Bohr–Sommerfeld Lagrangian submanifold).

The set of all transformations of this form $\mathbb{F}=\{ P(F),\, F \in C^{\infty}(M, \mathbb{R}) \}$ is an Abelian subgroup in $\operatorname{Aut} \mathbb{P} \Gamma (M, L)$, and it is not hard to find subspaces which are invariant with respect to this subgroup action. Let $D \subset M$ be a submanifold representing the homology class $\mathrm{P.D.}[\omega] \in H_{2n-2}(M, \mathbb{Z})$. Then the subset $\mathbb{P}(D)=\{ [\alpha] \mid (\alpha)_0=D \} \subset \mathbb{P} \Gamma (M, L)$ is of the desired type. Here, such $D$ is not obligated to be smooth, and it can consist of several components with multiplicities. The subset $\mathbb{P}(D)$ is not a projective subspace, since we require precise coincidence of the zero set of $\alpha$ and the submanifold $D$; however, it is not hard to see that this space is an affine subspace.

We call the group $\mathbb{F}$ a changing phase group, since its elements precisely change the phase factors of the sections.

The changing phase group contains one parameter subgroups of special type $\{ P(tF) \}$, which implies for a smooth function $F$ the corresponding vector field $\Theta_P(F)$ as the infinitesimal part of $P(tF)$, and by definition this vector field must preserve the Kahler structure on $\mathbb{P} \Gamma (M, L)$. Since $\mathbb{F}$ is commutative, the space of all such $\Theta_P(F)$ forms a distribution on the projective space $\mathbb{P} \Gamma (M, L)$. Clearly, this distribution is integrable. Let us find explicitly the corresponding integrable subsets.

First of all, for a section $\alpha \in \Gamma (M, L)$, the transformation $\alpha \mapsto e^{\imath F} \alpha$ preserves the pointwise norm $| \alpha |_h$, therefore, the projective version of the transformation $P(F)$ must preserve the real 1-form $d \ln | \alpha |_h$, which is correctly defined on the complement $M \setminus D$. Let us now recall a construction from [4] capable of delivering a natural representation of the component $\mathbb{P}(D)$ by the following rule: one associates with a class $[\alpha] \in \mathbb{P}(D)$ the complex valued 1-form $\rho(\alpha)=\nabla_a \alpha/\alpha$ (correctly defined on the complement $M \setminus D$) such that $\operatorname{Re}(\rho(\alpha))$ is exact, and $d \operatorname{Im} (\rho(\alpha))=2 \pi \omega$ (the real part is precisely $d \ln | \alpha |_h$). The key observation here is that this is a one-to-one correspondence: each complex-valued 1-form $\rho$ of this type on the complement $M \setminus D$ defines (uniquely, up to a constant) the corresponding section $\alpha$ which vanishes along $D$ (see [4]).

In these terms, the action of $P(F)$ looks quite simple: $\rho(P(F)(\alpha))= \rho(\alpha)+\imath\, d F$. Another key fact was established in [1]: in these terms, the SBS condition means simply that $\operatorname{Im} \rho(\alpha)|_S \equiv 0$.

This implies that the subset $\mathbb{P}(D)$ is an affine subspace associated with the complex vector space consisting of exact complex 1-forms on the complement $M \setminus D$. This affine space if sliced by the attachment $[\alpha] \mapsto \operatorname{Re} \rho(\alpha)$, and if we denote the fiber over a fixed element $\operatorname{Re} \rho(\alpha_0)$ by $\mathbb{P}^0 (D)$, then it is clear that the vector field $\Theta_P(F)$ must be tangent to the fiber at each point. This establishes the following result.

Note that $\mathbb{P}^0(D)$ is again an affine space, but now it is real one: it is associated with the space of real exact 1-forms on the complement. However, as pointed our above, for the SBS-constructions, only the imaginary parts of $\rho$-forms are important. We this have the following result.

Indeed, the SBS-condition speaks nothing about the real part $\operatorname{Re} \rho(\alpha)$, therefore, we can reduce $p$ to $p'$ by adding an appropriate exact real form to $\rho(\alpha)$.

Thus, to study $\mathcal{U}_{\mathrm{SBS}}(a)$, we can consider the components $\mathbb{P}^0(D)$ only, not the whole projective space $\mathbb{P} \Gamma (M, L)$. In particular, it would be interesting to find a certain universal rule for choosing the real part for $\operatorname{Im} \rho$ for a fixed submanifold $D \subset M$. In some cases, this choice can be done canonically: for example, if our symplectic manifold $(M, \omega)$ admits an integrable complex structure $I$ compatible with $\omega$, and if the submanifold $D$ is complex, then our $(M, \omega, I)$ is a complex Kahler manifold with Kahler metric of Hodge type. The choice of Hermitian connection $a_I$ in the orbit $\mathcal{O}(\omega)$ induces a holomorphic structure on $L$, and it is well known that the corresponding holomorphic section space $H^0(M_I, L)$ is a finite dimensional subspace in $\Gamma (M, L)$. Each holomorphic section $\alpha \in H^0(M_I, L)$ is defined (uniquely modulo constants) by the zero divisor $D_{\alpha}$ consisting of complex submanifolds with multiplicities. This gives us a realization of the projective space $\mathbb{P} H^0(M_I, L)$ as a complete linear system $|L|$.

Therefore, there is a finite dimensional family of affine subspaces $\{ \mathbb{P}(D)^0 \mid D \in |L| \}$ with marked points $p_D \in \mathbb{P}(D)^0$. Here, it is worth noting that such affine subspaces $\mathbb{P}^0(D)$ have empty intersection by definition.

§ 3. Deformations

In the previous section we studied possible transformations of the first elements in the pairs $(p, S)$; now we need to examine deformations of the second elements (Lagrangian submanifolds). In contrast to the first ones, they form neither projective nor affine spaces, therefore, instead of transformations, we will deal with deformations.

Indeed, if $S$ is a SBS submanifold with respect to $\alpha$, then, for any Darboux–Weinstein neighbourhood $\mathcal{O}_{\mathrm{DW}}(S) \subset M$ of the submanifold $S$, we have the corresponding 1-form $(1/(2\pi)) \operatorname{Im} \rho(\alpha)$ such that its differential is $\omega$ and its restriction to $S$ identically vanishes. Consider the canonical action 1-form $\alpha_{\mathrm{can}}$ on the neighbourhood $\mathcal{O}_{\mathrm{DW}}(S)$ and consider the difference $\alpha_{\mathrm{can}}-(1/(2\pi)) \operatorname{Im} \rho(\alpha)$. This difference is closed, and since the neighbourhood $\mathcal{O}_{\mathrm{DW}}(S)$ is contractible to $S$, along which this difference vanishes identically by the conditions of the proposition, there exists a smooth real function $F_0$ such that $d F_0=\alpha_{\mathrm{can}}- (1/(2\pi)) \operatorname{Im} \rho(\alpha)$ on $\mathcal{O}_{\mathrm{DW}}(S)$. Note that this function satisfies the condition $d F_0|_S \equiv 0$ by definition.

Now consider a new Hermitian prequantization connection $a_0$ such that $\nabla_{a_0} =\nabla_a-\imath\, d F_0$; from Proposition 1 it follows that the pair $([\alpha], S)$ belongs simultaneously to both the subsets $\mathcal{U}_{\mathrm{SBS}}(a)$, $\mathcal{U}_{\mathrm{SBS}}(a_0)$. At the same time, the local picture around the point $([\alpha], S)$ in the last subset is as follows. Since here $\alpha_{\mathrm{can}} \,{\equiv}\, (1/(2 \pi)) \operatorname{Im} \rho(\alpha)$, it follows that, for any small Hamiltonian deformation $S_{\delta}$ of the submanifold $S$, induced by smooth function $\phi\,{\in}\, C^{\infty}(S, \mathbb{R})$ as usual in the Darboux–Weinstein presentation (see [5]), we have $(1/(2 \pi)) \operatorname{Im} \rho(\alpha)|_{S_\delta}=d \pi^* \phi$ (here $\pi\colon \mathcal{O}_{\mathrm{DW}}(S) \to S$ is the canonical projection to the base transported from $T^*S \to S$). Therefore, if we consider the Hamiltonian deformation $S_{\delta}$ of our given submanifold $S$, as represented in the Darboux–Weinstein neighbourhood $\mathcal{O}_{\mathrm{DW}}(S)$ by the corresponding function $\phi \in C^{\infty}(S, \mathbb{R})$, then the restriction $(1/(2\pi))\operatorname{Im} \rho(\alpha)|_{S_{\delta}}$ is $d(\pi^* \phi)|_{S_{\delta}}$; furthermore, the real function $\pi^* \phi|_{S_{\delta}}$ can be extended to the Darboux–Weinstein neighbourhood $\mathcal{O}_{\mathrm{DW}}(S)$, and then, to the whole $M$ (this extension is denoted by $F_1 \in C^{\infty}(M, \mathbb{R})$). Note that the extension $F_1$ is supposed to agree with $\phi$ when restricted to $S_{\delta}$; therefore, such an extension always exists.

Now, consider the subset $\mathcal{U}_{\mathrm{SBS}}(a_0+\imath \, d F_1)$; evidently, the pair $([\alpha], S_{\delta})$ satisfies the SBS-condition with respect this prequantization Hermitian connection. Consequently, the same pair $([\alpha], S_{\delta})$ belongs to $\mathcal{U}_{\mathrm{SBS}}(a+ \imath \, d(F_0+ F_1))$, and if we denote the sum of the smooth functions by $\delta=F_0+F_1$, then the pair $(P(\delta) [\alpha], S_{\delta})$, which consists of the transformed first element and the deformed second element, must be contained in $\mathcal{U}_{\mathrm{SBS}}(a)$. This completes the proof of Proposition 5.

Note that during all the construction steps the first elements do not leave the affine space $\mathbb{P}(D)$.

Theorem ($\mathcal{B}_S$-covering theorem). Let $S_t,\, t \in [0,1]$, be a Hamiltonian isotopy of some Bohr–Sommerfeld Lagrangian submanifold $S_0$ such that $([\alpha_0], S_0) \in \mathcal{U}_{\mathrm{SBS}}(a)$ for a fixed prequantization connection $a$. Suppose that, for all $t \in [0,1]$, the intersection $S_t \cap D_{\alpha_0}$ is empty and each $S_t$ is smooth. Then there exists a corresponding family $([\alpha_t], S_t) \in \mathcal{U}_{\mathrm{SBS}}(a)$ such that all $[\alpha_t]$ are contained in the same affine space $\mathbb{P}^0(D_{\alpha_0})$.

Indeed, since Hamiltonian isotopies preserve the Bohr–Sommerfeld condition, each $S_t$ is Bohr–Sommerfeld; for each $t \in (0,1)$ the corresponding Lagrangian submanifold $S_t$ admits a Darboux–Weinstein neighbourhood $\mathcal{O}_{\mathrm{DW}}(S_t)$ such that every $S_{t'}$ for $t' \in (t-\varepsilon, t+ \varepsilon)$ belongs to this neighbourhood. The segment $[0,1]$ is compact, therefore, we can find a finite set of submanifolds $S_{t_i}$, $i=1, \dots, N$, such that the union of the corresponding Darboux–Weinstein neighbourhoods $\bigcup_{i=1}^N \mathcal{O}_{\mathrm{DW}}(S_{t_i})$ contains all $S_t$. Therefore, employing Proposition 5 several times, and passing through the segment, we find that the submanifold $S_1$ can be equipped with the corresponding class $[\alpha_1]$ in such a way that the conditions of $\mathcal{B}_S$-covering theorem are fulfilled, which completes the proof.

At the same time, we could exploit another method based on the stability of the SBS-condition with respect to the flow generated by a Hamiltonian vector field. Namely, let $F$ be a global function on $M$, $X_F$ be its Hamiltonian vector field, and $\phi^t_{X_F}$ be the corresponding flow generated by $X_F$. Then, for the pair $(p, S) \in \mathcal{U}_{\mathrm{SBS}}(a)$, there is a corresponding Hamiltonian deformation $(\phi^t_{X_F}(p), \phi^t_{X_F} (S))$. So, we have the following result.

For a proof, we note that the action of the flow $\phi^t_{X_F}$ on the pair $(p, S)$ admits a very simple and natural reformulation if, instead of the projectivization of the section $p=[\alpha]$, we consider the corresponding complex 1-form $\rho(\alpha)$, the zero set $D_{\alpha}$, and the Lagrangian submanifold $S$ itself. It is easy to see that if $S$ did not intersect $D$, then, under the action of the flow, the submanifolds $S_t=\phi^t_{X_F}(S)$ and $D_t =\phi^t_{X_F}(D)$ will remain disjoint, but then, for a fixed $D_t $, the section class can be reconstructed from the form $\rho_t$, and at the same time, the condition $\operatorname{Im}(\rho_t)|_{S_t} \equiv 0$ evidently remains to be satisfied for each $t$, which completes the proof.

§ 4. Definition of the moduli space

Consider a simply connected smooth compact (projective) algebraic variety $X$ with very ample line bundle $L \to X$ (which exists by the definition). We choose an appropriate Hermitian structure $h$ on $L$, and consider the corresponding Kahler form $\omega_h$ defined by the set of Kahler potentials $\psi_{\alpha} = -\ln | \alpha |_h$, which are induced by the holomorphic sections $\alpha \in H^0(X, L)$ on the complements $X \setminus D_{\alpha}$. At the same time, the choice of $h$ in the presence of a holomorphic structure on $L$ distinguishes the connection $a_I \in \mathcal{O}(\omega_h)$.

Thus, the choice of an appropriate $h$ leads us to the above situation: we have a prequantization quadruple $(M=X$, $\omega=\omega_h,\, I,\, L,\, a_I)$, and hence the constructions of SBS-geometry can be applied. In this circumstance, according to [2], we know the details of these constructions in the holomorphic situation, where, for each class of holomorphic sections $[\alpha] \in \mathbb{P} H^0(X, L) \subset \mathbb{P} \Gamma (X, L)$, the preimage $p_1^{-1}([\alpha] \in \mathcal{U}_{\mathrm{SBS}}(a_I))$ can be described explicitly as follows. We take the corresponding Kahler potential $\psi_{\alpha}$ on the complement $X \setminus D_{\alpha}$, consider its critical points $x_1, \dots, x_N$, and choose finite trajectories of the gradient flow induced only by the gradient vector field $\operatorname{grad} \psi_{\alpha}$ joining $x_i$ (at the same time, semi-infinite trajectories approach the pole at $D_{\alpha}$); then the union of all such finite trajectories forms the Weinstein skeleton $W(X \setminus D_{\alpha})$ of the complement, and then a smooth Bohr–Sommerfeld Lagrangian submanifold $S \subset X \setminus D_{\alpha}$ is special with respect to the class $[\alpha]$ if and only if $S$ is contained in $W(X \setminus D_{\alpha})$ (details can be found in [2]). On the other hand, according to [7], even in the simplest cases, the Weinstein skeleton $W(X \setminus D_{\alpha})$ cannot admit smooth top components, therefore, the proper preimage $p^{-1}_1([\alpha])$ for these cases is empty.

However, this holomorphic situation shows that if the dimension of the Weinstein skeleton is $n$, then it must contain non-trivial $n$-dimensional cycles, which can be considered as a homological base of special Bohr–Sommerfeld submanifolds. If we single out a primitive top class in $W(M \setminus D_{\alpha})$ for the generic holomorphic section, then, for any other generic section the same class must exist.

Let us return back to the previous arguments: the emptiness of the preimage follows from the fact that the above situation is too far from being generic — it is very special since the choices of the prequantization connection $a_I$ and of the holomorphic section $\alpha$ are related too strongly. If we consider a small perturbation $\delta\colon \mathbb{P} H^0(X, L) \to C^{\infty}(X \setminus D_{\alpha}, \mathbb{R})$ such that $[\alpha] \mapsto P(\delta([\alpha])) [\alpha]$ lies in the same affine space $\mathbb{P}(D_{\alpha})$, one can naturally expect that the situation turns out to be more generic and thus, the perturbed preimage $p_1^{-1}(P(\delta ([\alpha])))$ should be non-trivial.

The term “small” in relation to the perturbations consider below can be defined as follows: since each function $\delta([\alpha])$ 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 each $p$ with a certain fixed $\varepsilon$; moreover, we will consider only those perturbations which are non-constant in an appropriately small neighbourhood of the Weinstein skeleton $W(X \setminus D_{\alpha})$. If we denote by $\mathbb{P}H^0(X, L)_{\delta} \subset \mathbb{P} \Gamma (X, L)$ the corresponding deformation of our original finite dimensional projective space. Now this deformation easily implies the following result.

Indeed, as we have seen above, the projective space $\mathbb{P} H^0(X, L)$ is associated with the corresponding family $\{ \mathbb{P}^0 (D_{\alpha}) \}$ of pairwise non-intersecting affine subspaces transversal to our projective space. The transformations, which are induced by $\delta$, act along the “slices” $\mathbb{P}(D_{\alpha})$, which implies the smoothness, and from the fact that the deformation is sufficiently small, it follows that the resulting submanifold $\mathbb{P} H_0(X, L)_{\delta}$ remains symplectic.

Let us come back to the above pair $(X, L)$. For an appropriate Hermitian structure $h$, we give the following definition.

The dependence on the Hermitian structure $h$ was discussed in detail in [2]; our main goal at present is to show that the geometry of the moduli space does not depend on the choice of small perturbations (in what follows, we denote it by $\mathcal{M}_{\mathrm{SBS}}$ for brevity if the topological type and the homology class are clear from the context).

The proof of correctness of Definition 1, which is the essence of Proposition 8, is based on the constructions presented in the subsequent section, where we will prove that the above space $\mathcal{M}_{\mathrm{SBS}}$ is naturally isomorphic to a certain universal object which does not depend on perturbations (namely, to the stable component of the moduli space of $D$-exact Lagrangian submanifolds, as introduced in [3]).

Remark. A direct proof of Proposition 8 is possible and important, however it would be much more interesting to establish a fact which can be naturally called the $\mathbb{P}$-covering theorem. Above, we have proved the $\mathcal{B}_S$-covering theorem for the case where one parameter families of elements from the moduli space $\mathcal{B}_S$ is covered by natural deformations in $\mathcal{U}_{\mathrm{SBS}}(a)$. Now, we consider deformations of other elements from pairs belonging to the projective space $\mathbb{P} \Gamma (X, L)$: in this case, the situation looks more simple, since the subspace $\mathbb{P}(D_{\alpha})$ is affine, and, therefore, every two elements $p_1, p_2 \in \mathbb{P} (D_{\alpha})$ can be joined by a corresponding segment. Indeed, in the presence of the marked point $[\alpha]$, any other element $p_i \in \mathbb{P}(D_{\alpha})$ is uniquely represented by the 1-form $\operatorname{Im} \rho(\alpha)+d F$, where $F$ is a certain smooth function on $X$. Therefore, the segment with end-points at $p_1$ and $p_2$ is presented as the family $\operatorname{Im} \rho(\alpha)+t \, d F_1+(1-t) \, d F_2$; but for this deformation a possible $\mathbb{P}$-covering theorem cannot be true since if we take $F_2=-F_1$, then at the middle point $p=[\alpha]$ a discontinuity appears. At the same time, the space of possible deformation is huge: in the space of exact forms $d \Omega^0_X$ one can take any generic path with end-points at $d F_1$ and $d F_2$, and, for this generic path, such a theorem should be correct. However, the last version is also not correct, since the preimages $p^{-1}_1([\alpha'])$ can be presented by pairs $([\alpha'], S_i)$, where the second elements (Lagrangian submanifolds) are not deformation equivalent (for example, they can represent different homology classes in $H_n(W(X \setminus D_{\alpha}), \mathbb{Z})$), which implies non-existence of a covering path in $\mathcal{U}_{\mathrm{SBS}}(a)$. And if one, says, requires an additional condition that the second elements should be Hamiltonian equivalent themselves, then no new theorem would be required — the above $\mathcal{B}_S$-covering theorem would suffice.

§ 5. Exact Lagrangian submanifolds

In [3], for the same situation (an algebraic variety $X$ and a very ample line bundle $L \to X$), the author of the present paper constructed a moduli space of $D$-exact Lagrangian submanifolds, which was denoted by $\widetilde{\mathcal{M}}_{\mathrm{SBS}}$, and then, in this moduli space, a certain stable component $\mathcal{M}^{\mathrm{st}} \subset \widetilde{\mathcal{M}}_{\mathrm{SBS}}$ was singled out. The construction was augmented with some arguments supporting the use of the abbreviation SBS. However, no precise relations between these in principal different construction were revealed. Below, we will establish this relationship.

Let us biefly recall the main constructions: for a simply connected algebraic variety $X$ and a very ample line bundle $L \to X$, we choose an appropriate Hermitian structure‘$h$ generating on $X$ the corresponding symplectic form $\omega_h$. Then with element $p=[\alpha] \in \mathbb{P} H^0(X, L)$ we associate the set of $D$-exact smooth Lagrangian submanifolds $\{ S \mid S \subset X \setminus D_{\alpha} \}$ of fixed topological type $\operatorname{top} S$ and of homology class $[S] \in H_n(X, \mathbb{Z})$, which is non-trivial in the group $H_n(X \setminus D_{\alpha}, \mathbb{Z})$; then we factorize this set with respect to Hamiltonian isotopies of the complement $X \setminus D_{\alpha}$, which gives us a discrete set, and at the last step, we globalize the construction over the whole $\mathbb{P} H^0(X, L)$, which at the end gives us the space $\widetilde{\mathcal{M}}_{\mathrm{SBS}}$. In [3], it was shown that this space is an open smooth Kahler manifold covering an open subset in $\mathbb{P} H^0(X, L)$ without ramification. The notion of $D$-exactness coincides, in the case, with the standard one: a Lagrangian submanifold is said to be exact if the restriction $\operatorname{Im} \rho(\alpha)|_S$ is exact; evidently, this property, for a fixed $S$, does not depend on the choice of a concrete element from $\mathbb{P}(D_{\alpha})$. Our notion of $D$-exactness was introduced in [3] by the following reason: in these terms, stability of the exactness condition with respect to deformations of $D_{\alpha}$ is evident, namely if a certain $S$ is exact with respect to $[\alpha] \in \mathbb{P} H^0(X, L)$ and does not intersect the zero divisor $D_{\alpha}$, then, for any deformation $[\alpha']$, the submanifold $S$ remains to be exact until $D_{\alpha'}$ touches $S$. Below, we will not use the deformations of this sort, so we will exploit the standard definition of exactness.

Further in [3], a certain component in the constructed moduli space was singled out: we say that a cycle $\Delta$ contained in the Weinstein skeleton $W(X \setminus D_{\alpha})$ admits a Bohr–Sommerfeld resolution if there exists a homotopy $\{ S_t \}$, $t \in [0,1]$, such that $S_0=\Delta$, and, for each positive value of $t \in (0,1]$, the corresponding family element $S_t \subset X \setminus D_{\alpha}$ is a smooth Lagrangian Bohr–Sommerfeld submanifold. By definition, such $S_t$’s represent some fixed non-trivial class in $H_n(X \setminus D_{\alpha}, \mathbb{Z})$; moreover, stability of the exactness property implies that all these $S_t$ are exact. Therefore, from the moduli space $\widetilde{\mathcal{M}}_{\mathrm{SBS}}$ one can single out a stable component $\mathcal{M}^{\mathrm{st}}$, which is presented by the classes realizing such Bohr–Sommerfeld resolutions.

Since the space $\mathcal{M}^{\mathrm{st}}$ is fibred over $\mathbb{P} H^0(M_I, L)$, and the manifold $\mathcal{M}_{\mathrm{SBS}}$ is defined as the preimage over a small deformation of the same projective space, it is natural to consider an arbitrary element $[\alpha] \in \mathbb{P} H^0(M_I, L)$ and to show that there is a certain correspondence between the fiber over it in the first case and the fiber over its small deformation in the first case. If a correspondence will be established for an arbitrary $[\alpha]$, then the conclusion of the theorem should follow in view of the definitions of the moduli spaces via globalization of the local picture over a fixed $[\alpha]$ to the whole $\mathbb{P} H^0(M_I, L)$.

The identification scheme for the moduli space is based on the following relations: for a fixed $[\alpha]$, the choice of a perturbation corresponds to a certain deformation of the connection instead of the section, so we deform the connection $a_I \mapsto a=a_I+\imath \, d \delta$, leaving unchanged $[\alpha]$; at the same time, we know that by Proposition 2 all $\mathcal{U}_{\mathrm{SBS}}(a)$ are isomorphic to each other, thus, if, for some $a$ different from $a_I$ the element $[\alpha]$ underlies of a non-trivial preimage, then the same is true for all generic $a$; the exactness of Lagrangian submanifold is independent of the choice of a prequantization connection, and so, is independent of the perturbation as well.

First of all, we note that if the Weinstein skeleton $W(X \setminus D_{\alpha})$ contains as a component some smooth Lagrangian submanifold $S_{\alpha}$, then, on the one hand, we do not need to consider small perturbations of $[\alpha]$ for evaluation of the proper preimage $p_1^{-1}([\alpha])$, since it already exists; on the other hand, in this situation, for finding the corresponding point in $\mathcal{M}^{\mathrm{st}}$ it is enough to take the constant homotopy $\Delta S_{\alpha} \equiv S_t$. Therefore, for this case, we have the identity for the elements of the moduli space.

Now, assume that we have an $n$-dimensional closed cycle $\Delta \subset W(X \setminus D_{\alpha})$ which is not a smooth Lagrangian submanifold. If, for generic appropriate small perturbations $\delta_i$, one has a family of the corresponding Bohr–Sommerfeld Lagrangian submanifolds $S_i$, which are the elements of the same moduli space $\mathcal{B}_S$, and, consequently, they have the same topological type, then one can “span” a homotopy $S_t$, on these set starting form $S_0=\Delta$. In [3], it was explained in details why these Bohr–Sommerfeld Lagrangian submanifolds are exact; on the other hand, the dimension of the CW-complex $W(X \setminus D_{\alpha})$ is at most $n$, therefore, it cannot contain another cycle which presents the same homology class in $H_n(X \setminus D_{\alpha}, \mathbb{Z})$.

Note that the same closed cycle $\Delta$ can admit topologically different Bohr–Sommerfeld resolutions, but, in our construction, we are restricted by the originally fixed topological type of Lagrangian submanifolds.

On the other hand, if there exists a Bohr–Sommerfeld homotopy $S_t$ with a starting element $S_0=\Delta \subset W(X \setminus D_{\alpha})$, then each $S_t$ corresponds to some small perturbation $\delta_t$: indeed, the restriction $\rho(\alpha)$ to $S_t$ must be the exact 1-form $d f_t$, and the function $f_t$ can be extended to a function $F_{\delta_t}$ supported near $W(X \setminus D_{\alpha})$ since $S_t$ is sufficiently close to $W(X \setminus D_{\alpha})$. Using local Hamiltonian deformations of $S_t$, we get sufficiently dense covering of the space of possible $\delta$-perturbations of our given element $[\alpha]$, which implies non-triviality of the preimage $p_1^{-1}$ for a generic perturbation.

Thus, the theorem has a simple geometrical meaning: Bohr–Sommerfeld resolution of an $n$-cycle $\Delta \subset W(X \setminus D_{\alpha})$ can be identified either with $\delta$-perturbations of the first elements or with Hamiltonian isotopies of the second elements in the pairs $(p_t, S_t) \in \mathcal{U}_{\mathrm{SBS}}$.

Remark. At this point, the following natural question arises: why we need to consider the constructions of special Bohr–Sommerfeld geometry while we already have a good definition for the moduli space from [3]? The main argument supporting for the use of the SBS geometry can be derived from § 1, where we presented the construction of a universal bundle $\mathcal{L} \to \mathcal{U}_{\mathrm{SBS}}(a)$. By definition (see § 4), the moduli space $\mathcal{M}_{\mathrm{SBS}}$ is embedded into $\mathcal{U}_{\mathrm{SBS}}(a_I)$; it follows that the natural restriction $\mathcal{L}_{\delta} \to \mathcal{M}_{\mathrm{SBS}}$ is defined up to the choice of an appropriate small perturbation $\delta$. At the same time, it is not hard to see that topologically the restriction $\mathcal{L}_{\delta}$ does not depend on the choice of $\delta$, which gives us not just a manifold $\mathcal{M}_{\mathrm{SBS}}$ (as it is in the case of $\mathcal{M}^{\mathrm{st}}$), but a “manifold$+$bundle” pair. Variations $\delta$ for this bundle correspond to those of Hermitian structure, since it can be reformulated in terms of variations of the based connection $a_I$, and, according to the above, correspond to variations of covariantly constant sections $\sigma_S$, which correspond to variations of the local basis around the given point, what, in turn, correspond to variations of the corresponding Hermitian structure. On the other hand, in [3], a conjecture was put forward to the effect that the stable component $\mathcal{M}^{\mathrm{st}}$ of the moduli space of D-exact Lagrangian submanifolds is not just a Kahler manifold, but has the form of an “algebraic variety minus ample divisor”. Namely, the presence of an ample line bundle distinguishes the algebraic case from the more general Kahler one, therefore, the conjecture can be attacked in terms of the pair $\mathcal{L} \to \mathcal{M}_{\mathrm{SBS}}$: if, fixing an appropriate Kahler structure on the base, we will find a Hermitian connection on $\mathcal{L}$ with the curvature form proportional to the Kahler form and a section which does not vanish on the base $\mathcal{M}_{\mathrm{SBS}}$, then the desired algebraic structure would be constructed. A realization of this programme requires inventions of many technical details starting with the questions about connections on $\mathcal{L}$ slightly touched in § 1, and even if the conjecture would be found to fail, then ar least the developed machinery should be useful in the studies of differential geometry of the moduli space $\mathcal{B}_S$ with possible applications in the constructions of Geometric quantization.

§ 6. Weinstein structures and Eliashberg conjectures

The special Bohr–Sommerfeld geometry is closely related to the theory of Weinstein structures. Recall (see [6]) that a vector field $\lambda$ on an open symplectic manifold $M \setminus D_{\alpha}$ with symplectic form $\omega$ is called Liouville if the Lie derivative $\mathcal{L}_{\lambda} \omega=\omega$. As we have seen above, any regular section $\alpha \in \Gamma (M, L)$ with zero set $D_{\alpha}$ represented by the combination of $(2n-2)$-dimensional components with multiplicities, induces the corresponding Liouville vector field $\lambda_{\alpha}=\omega^{-1}(\operatorname{Im} \rho(\alpha))$. This field depends only on the projectivization class $[\alpha] \in \mathbb{P}\Gamma (M, L)$. On the other hand, we have the real function $\psi_{\alpha}=-\ln | \alpha |_h$ with a pole along $D_{\alpha}$. Here, we note that this case does not cover all the space of Liouville vector fields on $M \setminus D_{\alpha}$: since $H^1(M \setminus D_{\alpha}, \mathbb{Z})$ is non-trivial (it follows from the topological non-triviality of the prequantization bundle), it follows that to $\operatorname{Im} \rho(\alpha)$ one can add any closed non-exact 1-form and apply $\omega^{-1}$ to this sum. Therefore, the Bohr–Sommerfeld geometry is related to (but does not cover) the geometry of Liouville vector fields. Below, we discuss only the case of Liouville fields coming from the prequantization bundle sections.

For any Liouville vector field $\lambda$, the core $\operatorname{Core}(\lambda)$ is defined as the set formed by finite trajectories of the flow $\Phi^t_{\lambda}$; thus, $\Phi^t_{\lambda}(\operatorname{Core}(\lambda))=\operatorname{Core}(\lambda)$ for each $t$. Consequently, the core $\operatorname{Core}(\lambda_{\alpha}) \subset M \setminus D_{\alpha}$ defined by a regular section $\alpha$ can be characterized by the following properties: at any smooth point $p \in \operatorname{Core}(\lambda_{\alpha})$ the vector field $\lambda_{\alpha}$ is tangent to $\operatorname{Core}(\lambda_{\alpha})$, the core $\operatorname{Core}(\lambda_{\alpha})$ is stable with respect to the flow $\Phi^t_{\lambda_{\alpha}}$, the core $\operatorname{Core}(\lambda_{\alpha})$ does not intersect $D_{\alpha}$.

Hence, for a Liouville vector field $\lambda_{\alpha}$ defined by a regular section $\alpha \in \Gamma (M, L)$ we note that if a smooth Lagrangian submanifold $S$ is SBS with respect to $[\alpha]$, then $S$ must be contained in $\operatorname{Core}(\lambda_{\alpha})$. However, it is known (see [8]) that the components of $\operatorname{Core}(\lambda)$ can have bigger dimension than $n$. The strict bound takes place if a given Liouville vector field $\lambda$ is gradient-like for some smooth function $\phi$, which means that there exists such a compatible Riemann metric $g$ such that

Basic examples of Weinstein structures come from complex geometry: if there exists an almost complex structure $I$ such that our given section $\alpha \in \Gamma (M, L)$ is pseudoholomorphic with respect to $I$, then the section generates a Weinstein structure on the complement $M \setminus D_{\alpha}$: a desired function $\phi$ is just $\psi_{\alpha}=-\ln | \alpha |_h$, and the corresponding vector field $\lambda_{\alpha}$ is precisely a gradient field for this function with respect to Riemann metric $g$, reconstructed from $\omega$ and $I$. However, as we have seen above, in the case of integrable $I$, the corresponding skeleton is non-smooth and does not admit smooth components. On the other hand, each closed $n$-dimensional cycle in $W(M \setminus D_{\alpha})$ represents a non-trivial homology class in $H_n(M \setminus D_{\alpha}, \mathbb{Z})$.

We mention the following natural problem in the framework of Weinstein structures: is it possible to find some more generic property of the prequantization bundle sections than pseudoholomorphicity, which would automatically imply that the corresponding Liouville vector field $\lambda_{\alpha}$ can be completed to a pair which defines a certain Weinstein structure on $ M \setminus D_{\alpha}$?

Another set of problems is concerned with properties of Lagrangian submanifolds in $M\,{\setminus}\, D_{\alpha}$ for the case where a given $\lambda_{\alpha}$ admits an appropriate function $\phi$. This set contains the known Eliashberg conjectures on exact Lagrangian submanifolds. We formulate a special version of these conjectures for our mostly interesting case of compact Lagrangian submanifolds; the complete version of the Eliashberg conjectures cam be found in [8].

As already mentioned in the previous section, a Lagrangian submanifold $S \subset M \setminus D_{\alpha}$ is called exact if the restriction $\omega^{-1}(\lambda_{\alpha})|_S$ is an exact form. Suppose that a spair $(\phi, \lambda_{\alpha})$ defines a Weinstein structure on $M \setminus D_{\alpha}$ and let $S \subset M \setminus D_{\alpha}$ be a smooth compact Lagrangian submanifold. We raise the following conjecture.

The second conjecture is equivalent to saying that $S$ is contained in the Weinstein skeleton defined by $(\phi', \lambda')$, which implies the first conjecture. Both these conjectures were formulated by Ya. Eliashberg (see [8]) and hence are called Eliashberg conjectures.

In essence, both these conjectures can be subsumed into the special Bohr–Sommerfeld geometry. As it was shown in § 5, the exactness condition for a Lagrangian submanifold $S \subset M \setminus D_{\alpha}$ implies the Bohr–Sommerfeld condition; moreover, we can realize the exactness condition in geometrical terms of the moduli space $\mathcal{B}_S$.

Namely, let $S \subset M \setminus D_{\alpha}$ be a smooth exact Lagrangian submanifold with respect to a Liouville vector field $\lambda_{\alpha}$. Consider the moduli space $\mathcal{B}_S$ of Bohr–Sommerfeld Lagrangian submanifolds of fixed topological type $S$ as a submanifold in whole $M$. Then the zero set $D_{\alpha}$ cuts the determinant set

The Liouville vector field $\lambda_{\alpha}$ induces a vector field $\Theta(\alpha)$ on the component $\mathcal{B}_S^i$ by the following rule: the restriction $\omega^{-1}(\lambda_{\alpha})|_S$ is an exact 1-form of the form $d f$, $f \in C^{\infty}(S, \mathbb{R})$, and any such exact form is a tangent vector from $T_{[S]} \mathcal{B}_S$ (see [5]). For the tangent vectors of this type, the norm is naturally defined by

For our construction, an affirmative answer to this problem should mean that the moduli space of exact Lagrangian submanifold coincides with its stable component (for definitions, see § 5). Indeed, since by definition, $S$ and $S'$ represent elements of the same connected component, they can be joined by a segment on $\mathcal{B}_S^i$, and, therefore, by a Hamiltonian isotopy. On the other hand, the norm $|\Theta(\alpha)([S])|$ measures how is $S$ far from $W(M \setminus D_{\alpha})$, and if Problem 1 has a positive answer, then the chain $S, S', \dots$ defines a homotopy which approaches a certain limit cycle, which must be a component in the Weinstein skeleton $W(M \setminus D_{\alpha})$. Therefore, we would find an element from $\mathcal{M}^{\mathrm{st}}$.

On the other hand, this should imply that $S$ is connected by a Hamiltonian deformation to a limit cycle $\Delta \subset W(M \setminus D_{\alpha})$, which by the topological arguments must be non-trivial, and thus, this would prove Conjecture 1 in our special case.

In view of the transformations, described in § 2, it is natural to ask about local variations of Weinstein structures. Namely, suppose that a Liouville vector field $\lambda_{\alpha_1}$ induced by a section $\alpha_1$ is pseudoholomorphic with respect certain compatible almost complex structure $I$ on the complement $M \setminus D_{\alpha_1}$; so, if one takes $\phi=\psi_{\alpha_1}$, then the pair $(\phi, \lambda_{\alpha_1})$ defines a Weinstein structure. Hence we can consider variations of the second element in the pair, which result in a Weinstein structure again. For example, an appropriate variation can be defined as $\lambda_t=\lambda_{\alpha_1}+(t-1) X_{\phi}$, $t \in [1,+\infty)$, where, as usual, $X_{\phi}$ denotes a Hamiltonian vector field of the function $\phi$ (moreover, one can take any composite function of $\phi$). It is easily seen that, for each $t \geqslant 1$, there exists a positive constant $C_t$ such that the pair $(\phi, \lambda_t)$ satisfies condition (2) with respect to the same metric. Indeed, $d \phi(\lambda_t) \equiv d (\lambda_1)$ and $| \lambda_t |_g^2=(1+t^2) | \lambda_1 |^2_g$, since $X_{\phi}$ and $\lambda_1$ are related by the operator of our almost complex structure, being the Hamiltonian vector field and the gradient vector field of the same function. So, one can take $C_t= C(1+t^2)$.

Geometrically, this means that we deform the components of our given Weinstein structure along the level sets of the potential $\phi$ by fixing its critical points, which are unchanged under this deformation.

On the other hand, condition (2) implies that the Liouville vector field $\lambda$, which lies in some Weinstein structure, has many critical points, since if $x \in M \setminus D$ is a critical point of the corresponding function $\phi$, and, at the same time, it is not a singular point for $\lambda$, we immediately arrive at a contradiction to condition (2). Hence we have the inclusion of the critical point sets $\{ \operatorname{Crit} \phi \} \subseteqq \{ \operatorname{sing} \lambda \}$. Note that if we fix a set of points $\{p_1, \dots, p_l \} \subset M \setminus D$, then this choice defines a subalgebra in the algebra $C^{\infty}(M, \mathbb{R})$ consisting of smooth functions $F$ such that $\operatorname{Crit} F \supseteqq \{p_1, \dots, p_l \}$. Indeed, if $d F_1 (p_i)=d F_2 (p_i)=0 $, then $d(F_1+c F_2)(p_i)=0$ and $d (F_1 \cdot F_2) (p)=0$. Consequently, for a given Weinstein structure $(\phi, \lambda)$ we have the following variation space: the zero set of the vector field defines the subalgebra $\mathcal{A}(\lambda) \subset C^{\infty}(M, \mathbb{R})$, and one can find a sufficiently small neighbourhood of zero in this subalgebra such that, for any sufficiently small deformation $\delta \in \mathcal{A}(\lambda)$, the pair $(\phi+\delta, \lambda)$ again defines a Weinstein structure.

Thus, a possible strategy for passing from above Problem 1 to Conjecture 2 is as follows: first, we deform $S$ using Hamiltonian deformations such that the corresponding $S'$ should be as close as possible to the Weinstein skeleton $W(M \setminus D_{\alpha})$, and second, we find an appropriate deformation of the Weinstein structure itself so that this smooth submanifold $S'$ turns to be a component of the new Weinstein skeleton.

This approach involves the introduction of a certain distance function on the space of exact Lagrangian submanifold, for otherwise there is no meaning in proximity of $S'$ to $W(M \setminus D_{\alpha})$. It turns out that the above notion of a norm on the vector field on the moduli space $\mathcal{B}_S$ or on its component can be naturally generalized to the case of smooth compact exact Lagrangian submanifolds on a Weinstein domain $M \setminus D_{\alpha}$.

Suppose, as above, that a pair $(\phi, \lambda)$ defines a Weinstein structure on an open symplectic manifold $M \setminus D_{\alpha}$ (here $\alpha$ is not necessary related to $\phi$ or $\lambda$). For any compact smooth Lagrangian submanifold $S \subset M \setminus D_{\alpha}$, we can define a distance, since, by definition, the restriction $\omega^{-1}(\lambda)|_S=d f_{\lambda}$, where $f_{\lambda} \in C^{\infty}(S, \mathbb{R})$, is uniquely defined up to an addition of a constant. So, we arrive at the following definition.

It is easily seen that $N_{\lambda}(S) \in \mathbb{R}_{\geqslant 0}$; moreover, $N_{\lambda}(S)=0$ if and only if $S$ is a component of the Weinstein skeleton $W(M \setminus D_{\alpha})$. Indeed, $N_{\lambda}(S)=0 $ should imply $\omega^{-1}(\lambda)|_S \equiv 0$, therefore, $\lambda$ must be tangent to $S$ at each point, which, by the main property of the Weinstein skeleton, implies the required result. This explains the name of the function $N_{\lambda}(S)$.

As shown above, each smooth exact Lagrangian submanifold can be included in the corresponding component $\mathcal{B}_S^i \subset \mathcal{B}_S$ consisting of Hamiltonian equivalent on the complement $M \setminus D_{\alpha}$ of Bohr–Sommerfeld Lagrangian submanifolds. Therefore, the distance function is naturally defined on the entire component

Since the proposition is purely local, it is sufficient to consider the picture on a Darboux–Weinstein neighbourhood only of an arbitrary Lagrangian submanifold $S \subset M \setminus D_{\alpha}$ such that $[S] \in \mathcal{B}_S^i$. By the Darboux–Weinstein theorem, the neighbourhood $\mathcal{O}_{\mathrm{DW}}(S) \subset \mathcal{B}_S^i$ is symplectomorphic to some neighbourhood $\mathcal{O}_{\varepsilon}$ of the zero section in $T^*S$ taken with the standard symplectic form $\omega_{\mathrm{can}}=d a_{\mathrm{can}}$, where $a_{\mathrm{can}}$ is the canonical 1-form (details can be found in [5]). Let us transport the Liouville vector field $\lambda$ to $\mathcal{O}_{\varepsilon}$ and denote the corresponding field by $\lambda_0$. Evidently, the restriction of the function $N_{\lambda}|_{\mathcal{O}_{\mathrm{DW}}(S)}$ coincides with the function $N_0$ defined on exact 1-forms of type $d f$, $f \in C^{\infty}(S, \mathbb{R})$, by the following rule. First, the 1-form $\rho_0=\omega_{\mathrm{can}}^{-1}(\lambda_0)$ on $\mathcal{O}_{\varepsilon}$ is presented in the form

The transformation $A_{F_0}$ has many interesting properties. Let us show that $A_{F_0}$ is injective. Indeed, suppose that, for a pair of functions $f_1$, $f_2$, we have $A_{F_0}(f_1) \equiv A_{F_0}(f_2)$. If $d f_1=d f_2$, we immediately get a contradiction, since in this case, $f_2-f_1=\mathrm{const}$, and the restrictions $F_0(\Gamma(d f_i))$ are the same. Further, if $d f_1 \neq d f_2$, then the graphs $\Gamma(d f_i)$ must have at least two intersection points (at the global maximum and minimum of the difference $f_1-f_2$, respectively). Let us denote these points by $p_+, p_-\in \mathcal{O}_{\varepsilon}$ and connect them by paths $\gamma_i \subset \Gamma(d f_i)$ such that their projections to the zero set $\pi(\gamma_1)$, $\pi(\gamma_2)$ coincide. If $A_{F_0} (f_1) \equiv A_{F_0}(f_2)$, then the integrals $\int_{\pi(\gamma_i)} d A_{F_0}(f_i)$ are equal. But then

The image of the zero function $A_{F_0}(0)$ is the function $f_{\lambda}$ defined above. Let us formulate the following interesting problem.

In other words, is it true that, for any function sufficiently close to $f_{\lambda}$, the preimage with respect to the action of $A_{F_0}$ can be found? Essentially, the inverse operation $A_{F_0}^{-1}$ (if exists) should make possible the solution of some partial derivative equations, and then should show that the differential $d p_1$ is invertible, and this fact should help in the studies on special Bohr–Sommerfeld geometry. On the other hand, such an operator looks a bit similar (with respect to its functional) to the Maslov operator, and so, if one would succeed in establishing a more precise relation, then this would provide a generalization of the technique of semi-classical approximations in the general case of compact symplectic manifold. As a complementary problem, we can consider linearization of our non-linear transformation $A_{F_0}$ defined by linear operator of the form $f \mapsto L_0(f)=f+ d f (Y_0)$, where $Y_0$ is a smooth vector field on $S$ corresponding to the horizontal component of the Hamiltonian vector field $X_{F_0}$ at points of the zero section. On the compact manifold, the operator $L_0$ has trivial kernel and maps constants to constants, and so it is natural to expect that the image of this operator is everywhere dense, which should imply many useful corollaries.

Let us now come back to the proof of Proposition 10. To verify smoothness of the function $N_0$, consider the derivative of this function in a given direction $[S]+\delta \, d f $, where $f$ is a smooth function on $S$. The expression we need to find has the form

Now, let us show that the function $N_0$ can be increased of decreased by deformations of the zero section in $\mathcal{O}_{\varepsilon}$ if and only if $f_{\lambda} \neq \mathrm{const}$, which is equivalent to $N_0(0) \neq 0$. For this, we note that the Liouville vector field $\lambda_0$ acts on exact Lagrangian submanifold by Hamiltonian deformations. Indeed, the flow $\phi^t_{\lambda_0}$ acts on the 1-form $\omega^{-1}_{\mathrm{can}}(\lambda_0)$ by multiplication on $e^t$, therefore, if the restriction $\omega^{-1}_{\mathrm{can}}(\lambda_0)|_S$ is $f_{\lambda}$, then the restriction of the same 1-form on $\phi^t_{\lambda_0}(S)$ must be equal to $e^t f_{\lambda}$. Here, the norm $| e^t f_{\lambda} |_{\mathrm{sup}}$ is clearly different for $t \neq 0$ from the norm $| f_{\lambda} |_{\mathrm{sup}}$ if and only if the last one is not equal to zero.

Consider as $S$ the zero section in $\mathcal{O}_{\varepsilon}$. If $f_{\lambda}= \mathrm{const}$, then the Liouville vector field $\lambda_0$ is tangent to $S$, and so, $\phi^t_{\lambda_0}(S) \equiv S$. In the case where $f_{\lambda}$ is not constant, we can deform the zero section $S$ using the flow $\phi^t_{\lambda_0}$ with positive and negative $t$ (of sufficiently small absolute value). As a result, we get a pair of deformations $S_{\pm}$ (depending on the sign of $t$) such that $N_0(S_+)>N_0(S)$ and $N_0(S_-)< N_0(S)$, completing the proof of Proposition 10.

Proposition 10 is a local solution to Problem 1: by transporting the result under the identification by the Darboux–Weinstein symplectomorphism, we can claim that any exact (with respect to the Liouville vector field $\lambda$) Lagrangian submanifold $S$ can be deformed so that the norm of the restriction $\omega^{-1}(\lambda)$ decreases. The main question is how much this norm can be decreased? If it will be possible to find an effective bound (depending, say, on the topological type of $S$ and some properties of the Liouville vector field $\lambda$) for a possible decrease of the norm $| f_{\lambda} |_{\mathrm{sup}}$, then Problem 1 would be solved, and, consequently, we would get a desired result for testing the first of the two Eliashberg conjectures on exact Lagrangian submanifolds.

Citation in format AMSBIB

\Bibitem{Tyu23}

\by N.~A.~Tyurin

\paper Special Bohr--Sommerfeld geometry: variations

\jour Izv. RAN. Ser. Mat.

\yr 2023

\vol 87

\issue 3

\pages 184--205

\mathnet{http://mi.mathnet.ru/im9374}

\crossref{https://doi.org/10.4213/im9374}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4640919}

\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2023IzMat..87..595T}

\transl

\jour Izv. Math.

\yr 2023

\vol 87

\issue 3

\pages 595--615

\crossref{https://doi.org/10.4213/im9374e}

\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001063937600007}

\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85171833888}

Introduction