N. A. Tyurin, “Examples of Hamiltonian-minimal Lagrangian submanifolds in $\operatorname{Gr}(r, n)$”, Izv. Math., 89:2 (2025), 359

Abstract: We extend the A. E. Mironov's construction of (Hamiltonian)-minimal Lagrangian submanifolds to the case of an algebraic manifold that can be equipped with a Kahler–Einstein metric symmetric under the action of the torus $T^k$. As an application, we give examples of Hamiltonian-minimal Lagrangian submanifolds of the Grassmannian $\operatorname{Gr}(r, n)$.

Keywords: algebraic manifold, Grassmannian, Lagrangian submanifold, torus action, Hamiltonian minimality.

Funding agency	Grant number
National Research University Higher School of Economics
This study was prepared within the framework of the project “International academic cooperation” HSE University.

Received: 26.04.2024

Published: 12.05.2025

Bibliographic databases:

Document Type: Article

UDC: 514.763.424+514.763.337

MSC: Primary 14M99, 53D12; Secondary 14M15L

Language: English

Original paper language: Russian

Introduction

In 2004, Mironov [1] introduced the set of new examples of (Hamiltonian)-minimal Lagrangian submanifolds in $\mathbb{C}^n$ and $\mathbb{C} \mathbb{P}^n$. In particular, his work had substantially expanded the list of topological types that can be interpreted as Lagrangian submanifolds of the projective space. It also turned out that the methods of this work can potentially be generalized to the case of an algebraic manifold equipped with torus $T^k$ acting with Kahler isometries. The renewed interest in Lagrangian submanifolds of algebraic manifolds and in classification of Lagrangian submanifolds, has attracted the attention of the scientific community to Mironov’s method; eventually, in this way, natural generalizations to the case of an arbitrary algebraic manifold were obtained.

It is well known that the projective space $\mathbb{C} \mathbb{P}^n$ equipped with standard Fubini–Study metric serves as one of the most important examples of a compact symplectic manifold, with Kahler form $\omega_{FS}$ serving as a corresponding symplectic form. In this case, there are two natural possibilities for constructing Lagrangian submanifolds. The first one involves the existence of a compatible anti-holomorphic involution $\tau\colon \mathbb{C} \mathbb{P}^n \to \mathbb{C} \mathbb{P}^n$. The second one involves the torus $T^n$ acting on $\mathbb{C} \mathbb{P}^n$ with Kahler isometries, for which there exists of a full set of momentum maps $\mu_1, \dots, \mu_n$ such that, in an appropriate set of homogeneous coordinates $[z_0: \dots : z_n]$,

These two examples of Lagrangian submanifolds differ by the “degree of homogeneity” (or the dimension of the vector space of corresponding symmetries). We denote the first one by $S_0$, and the second one, by $S_n$. The natural question here is whether the gap between $0$ and $n$ can be filled by some other types of Lagrangian submanifolds. Mironov’s method provided an answer to this question: for any $k$ from $1$ to $n-1$, there exists a remarkably easy way to construct an example of a Lagrangian submanifold with the corresponding degree of homogeneity. Namely, we choose an appropriate set of values $c_1, \dots, c_k$ for the first $k$ momentum maps $\mu_1, \dots, \mu_k$, and put

Note that the cases we consider here are the most simple ones, since, for construction, one can also choose a sub-torus $T^k \subset T^n$ spanned by linear combinations of $\{\mu_i\}$ (this gradually expands the set of examples). However, even the above simple examples provide us with the set of $n+1$ Lagrangian submanifolds of $\mathbb{C} \mathbb{P}^n$ with symplectic form $\omega_{FS}$ that all have different topological types.

The above assertion was obtained in [1] as a side result, because, as was shown in [1], the constructed Lagrangian embeddings and immersions satisfy a very strong property of being (Hamiltonian)-minimal with respect to the standard Fubini–Study metric. Conditions of this kind are capable of singling out finite-dimensional subspaces in (typically) infinite-dimensional spaces of Lagrangian manifolds. The concept of minimality plays a very important role in different domains of differential geometry including symplectic geometry. In [2], Oh proposed a generalization of this concept called Hamiltonian minimality — by using this concept it proved possible to obtain approbate representatives in cases where there are no smooth minimal representatives. A minimal Lagrangian submanifold minimizes the Riemannian volume modulo all Lagrangian deformations, whereas a Hamiltonian-minimal Lagrangian submanifold has the same property but modulo all Hamiltonian deformations (that is, the deformations generated by Hamiltonian fields).

Usually, minimality problems are posed in the following situation: let $X$ be an algebraic manifold that can be equipped with a Kahler–Einstein metric $(g, I, \omega)$ satisfying $\operatorname{Ric}(g) = \lambda \omega$. Here, as usual, $\operatorname{Ric} (g)$ is the Ricci tensor of $g$, and $\omega$ is the corresponding Kahler form. For the canonical bundle $K \to X$, there exists a Hermitian connection $\nabla$ induced by Levi-Civita connection of the metric $g$. It is well known that the curvature of $\nabla$ is $2 \pi \imath \operatorname{Ric}(g)$. Therefore, the restriction of the pair $(K, \nabla)$ to an arbitrary Lagrangian submanifold $S \subset X$ is a flat linear bundle. This means that locally, for an arbitrary open neighbourhood $U \subset S$, there exists a covariantly constant section $\sigma$ of $(K, \nabla)|_U$ that has unit norm at any point. This section is defined uniquely up to the constant element from $U(1)$. Globally, such a section exists only in special cases; however, to define the Hamiltonian minimality it suffices to cover $S$ by an appropriate system of neighbourhoods and test the required properties locally over each $U$. The real linear bundle $\det TS$ is trivial over $U$, and so we choose its arbitrary trivialization $\xi$, that is, the top polyvector field that does not vanish on $U$. Since $S$ is Lagrangian, $\sigma(\xi)$ does not vanish on $U$. In addition, the function $\beta = \operatorname{Arg}(\sigma(\xi))$ (called the Lagrangian angle) is well defined. By construction, the Lagrangian angle does not depend on the choice of $\xi$, but depends on the choice of $\sigma$ up to a constant (a change of orientation on $U$ leads to a shift on $\pi$, which can be seen as a shift by a constant as well). The function $\beta$ may be not globally defined on $S$; however, its differential $m(S) = d \beta$ is well defined and coincides with the 1-form of the mean curvature of the Lagrangian submanifold $S$.

In these terms, the Lagrangian submanifold $S$ is minimal if $\beta = \mathrm{const}$ for any $U\subset S$. According to [2], $S$ is Hamiltonian minimal if, for any $U\subset S$, the function $\beta$ is harmonic with respect to the metric $g$ (equivalently, $S$ is Hamiltonian minimal if the form $m(S)$ of the mean curvature is harmonic, that is $d m(S) = d^* m(S) \equiv 0$). This point of view was used in [1]: it is known that the Fubini–Study metric features the required property of being Einsteinian, and so the required property is shown by verifying that the corresponding Lagrangian angles are locally harmonic.

Mironov’s method proved universal. In particular, it can be naturally extended to the case of an algebraic manifold $X$ equipped with Kahler metric invariant under the action of torus $T^k$ with moment maps $\mu_1, \dots, \mu_k$. The paper [3] introduces the construction called the Lagrangian mutation of a Lagrangian submanifold $S_0\subset X$ with respect to the action of $T^k$

The set of algebraic manifolds with such structures is quite broad, and so we get an opportunity to construct examples of Lagrangian submanifolds in many interesting cases. One of such examples is the Grassmann manifold $\operatorname{Gr}(r, n)$ of projective $r$-dimensional subspaces in $\mathbb{C} \mathbb{P}^n$. The action of $T^n$ on $\mathbb{C} \mathbb{P}^n$ induces naturally the $T^n$-action on the Grassmannian. Therefore, by choosing the Kahler structure on $\operatorname{Gr}(r, n)$ induced by Plücker embedding, one can easily describe the natural moment maps $\widetilde \mu_i$, of which each is generated by the moment map (1) and generates itself Kahler isometries. At the same time, the real submanifold $\operatorname{Gr}_{\mathbb{R}}(r, n)$ consisting of all $r$-dimensional real subspaces of $\mathbb{C} \mathbb{P}^n$ is naturally defined. Since $\operatorname{Gr}_{\mathbb{R}}(r, n)$ has a proper dimension, it can be used as $S_0$. Thus, by applying the Lagrangian mutations with respect to $T^k$ for $k = 1, \dots, n$, we get the list of potential candidates for the role of Lagrangian submanifolds. The main problem is to find out whether these subsets are smooth submanifolds for an appropriate choice of values of the moment maps $\widetilde \mu_1, \dots, \widetilde \mu_k$.

This problem was partially considered [4] and [5]. In [4], a way was found to choose a value for the moment map $\widetilde \mu_1$ in order to get a smooth Lagrangian submanifold $S_1$. In [5], for the case of Grassmannian $\operatorname{Gr}(1, n)$, appropriate values for $\widetilde \mu_1, \dots, \widetilde \mu_k$, $k< n$, were found with the corresponding smooth Lagrangian submanifolds $S_k$ whose topological type is a certain $T^{2k}$-bundle over $\operatorname{Gr}_{\mathbb{R}}(1, n-k)$.

However, so far, no question about Hamiltonian minimality of the constructed submanifolds was raised, even though the Grassmannian manifold $\operatorname{Gr}(r, n)$ admits a Kahler–Einstein metric, and hence is quite suitable for dealing with such questions. The present paper fills this gap. In § 1, we show that an extension of Mironov’s construction to an algebraic manifold with Kahler–Einstein metric invariant under $T^k$ gives Hamiltonian-minimal Lagrangian submanifolds. In § 2, we extend our construction from [5] from $\operatorname{Gr}(1, n)$ to the general case of the Grassmannian $\operatorname{Gr} (r, n)$, where $k = 1, \dots, n-r$. Together, these two results provide a list of Hamiltonian-minimal Lagrangian submanifolds of $\operatorname{Gr} (r, n)$.

§ 1. Hamiltonian-minimal Lagrangian submanifolds

Let $X$ be a projective variety of dimension $N$ with Kahler–Einstein metric $g$ such that the Ricci tensor $\operatorname{Ric}(g)$ is proportional to the Kahler form $\omega$. Also let $X$ be equipped with an anti-holomorphic involution $\tau\colon X \to X$ compatible with the Kahler structure and such that the real part $X_{\mathbb{R}} \subset X$ has maximal possible dimension $\dim_{\mathbb{R}} X_{\mathbb{R}} = \dim_{\mathbb{C}} X$. Next, suppose that on $(X, g)$ one has an action of torus $T^n$ by Kahler isometries, and one has a set $\mu_1, \dots, \mu_n$ of the moment maps such that $d \tau (X_{\mu_i}) = - X_{\mu_i}$ for each $i = 1, \dots, n$.

We choose an appropriate set of non-critical values $c_1, \dots, c_k$ of the moment maps $\mu_1, \dots, \mu_k$ such that the corresponding Lagrangian mutation

By definition, $X_{\mathbb{R}} \cap S_k = S_{\mathbb{R}}(c_1, \dots, c_k)$, and, since the context implies that certain non-critical values $c_1, \dots, c_k$ are chosen and fixed, and so, for brevity, in the proof the last submanifold will be denoted just as $S_{\mathbb{R}}$.

For an arbitrary point $p \in S_{\mathbb{R}}$, consider a neighborhood $U(p) \subset X$ of this point in the manifold $X$, and denote by $U_{\mathbb{R}}(p) \subset X_{\mathbb{R}}$ and $U_k(p) \subset S_k$, respectively, the corresponding neighborhoods of $p$ in $X_{\mathbb{R}}$ and $S_k$. Both $X_{\mathbb{R}}, S_k $ are Lagrangian, and hence the restrictions of $(K, \nabla)$, where $K \to X$ is the canonical bundle with connection $\nabla$ induced by the Levi-Civita connection to each of these submanifold are flat bundles. Over the open neighbourhoods $U_{\mathbb{R}}(p)$ and $U_k(p)$ one has the corresponding covariantly constant sections $\sigma_{\mathbb{R}}$ and $\sigma_k$, both having constant unit norms; and it is not hard to see that these sections can be chosen so that they would coincide under the intersection $U_{\mathbb{R}}(p) \cap U_k(p) \subset S_{\mathbb{R}}$.

Since $X_{\mathbb{R}} \subset X$ is a minimal Lagrangian submanifold, the Lagrangian angle $\beta$ is constant along it. This means that, for all points from $U_{\mathbb{R}}(p) \subset X_{\mathbb{R}}$, the following condition holds: if one substitutes into the top holomorphic form $\sigma_{\mathbb{R}}$ the top polyvector field $\xi_{\mathbb{R}}$ generated by a set of vector fields tangent to $X_{\mathbb{R}}$, then the result $\sigma_{\mathbb{R}}(\xi_{\mathbb{R}})$ has constant argument. Without loss of generality, the set of the tangent vector fields at points of $U_{\mathbb{R}}(p) \cap S_{\mathbb{R}}$ can be chosen as follows: $\langle\operatorname{grad} \mu_1, \dots, \operatorname{grad} \mu_k, v_{k+1}, \dots, v_N\rangle$, where $v_j$ form an orthonormal basis in the space of tangent vectors to $S_{\mathbb{R}}$. Indeed, by definition $S_{\mathbb{R}}$ is given in $X_{\mathbb{R}}$ by the conditions $\{\mu_i = c_i \}$. Therefore, at each point, the gradients of the moment maps $\mu_i$ are orthogonal to $T S_{\mathbb{R}}$; on the other hand, the gradients are linearly independent at each point, since by the condition the Hamiltonian vector fields $\{ X_{\mu_i} \}$ induced the $T^k$-action at each point are linearly independent.

$$ \begin{equation*} \xi_{\mathbb{R}} = \operatorname{grad} \mu_1 \wedge \dots \wedge \operatorname{grad} \mu_k \wedge v_{k+1} \wedge \dots \wedge v_N, \end{equation*} \notag $$

We now note that, by construction, the tangent space to $S_k$ at each point in $S_{\mathbb{R}}$ is spanned by the Hamiltonian vector fields $X_{\mu_1}, \dots, X_{\mu_k}$ together with the tangent vectors to $S_{\mathbb{R}}$, the latter can be chosen as values of the vector fields $v_{k+1}, \dots, v_N$. But since the gradient and the Hamiltonian vector field of a function are related by the complex structure operator, at points of $U_{\mathbb{R}}(p) \cap S_{\mathbb{R}}$ we can choose the top vector field for the submanifold $S_k$ in the form

$$ \begin{equation*} \begin{aligned} \, \xi_k &= X_{\mu_1} \wedge \dots \wedge X_{\mu_k} \wedge v_{k+1} \wedge \dots \wedge v_N \\ &= J_X(\operatorname{grad} \mu_1) \wedge \dots \wedge J_X(\operatorname{grad} \mu_k) \wedge v_{k+1} \wedge \dots \wedge v_N, \end{aligned} \end{equation*} \notag $$

The above arguments remain true for a neighbourhood of any point $p \in S_{\mathbb{R}}$, and so we have the first step of our proof: the Lagrangian angle $\beta_k$ calculated over the Lagrangian submanifold $S_k$ is constant along the submanifold $S_{\mathbb{R}}$.

Thus we get the first key fact of the proof: the kernel of the mean curvature form $m(S_k) \in \Omega^1_{S_k}$ contains $T_p S_{\mathbb{R}}$ at each point $p \in S_{\mathbb{R}}$. Indeed, the mean curvature $m(S_k)$ is the differential of the Lagrangian angle function, which implies the claim.

Next, both the metric $g$ and the Lagrangian submanifold $S_k$ are invariant under the Hamiltonian action of $X_{\mu_i}$, and so the mean curvature $m(S_k)$ is also invariant. Consequently, $\mathcal L _{X_{\mu_i}} m(S_k) = 0$. But the form $m(S_k)$ is closed, and, therefore, by the Cartan formula

Remark. It is easily seen that Theorem 1 remains valid if, instead of $X_{\mathbb{R}}$, one considers any minimal Lagrangian submanifold $S_0 \subset X$ transversal to the toric action. Indeed, the relations for the top polyvector fields $\xi_0$ on $S_0$ and $\xi_k$ on $S_k$ have exactly the same form, and so the further analysis repeats verbatim. The following more general question arises: if we start with a given Hamiltonian-minimal submanifold $S_0$, what can be said about its Lagrangian mutations? Is it Hamiltonian-minimal again, or one should impose certain additional conditions on $S_0$ to have the same answer?

§ 2. The case of Grassmannian $\operatorname{Gr}(r, n)$

A natural candidate to which Theorem 1 can be applied is the Grassmann variety $\operatorname{Gr}(r, n)$ of projective $r$-dimensional subspaces in $\mathbb{C} \mathbb{P}^n$ (details can be found, for example, in [6]): the Plücker embedding induces a Kahler–Einstein metric on it, the standard anti holomorphic structure and toric action on $\mathbb{C} \mathbb{P}^n$ induce the corresponding involution and $T^n$-action on the Grassmannian such that the real part $\operatorname{Gr}_{\mathbb{R}} (r, n) \subset \operatorname{Gr}(r, n)$ formed by real $r$-dimensional projective subspaces has maximal possible dimension as in Theorem 1. The corresponding moment maps $\widetilde \mu_i$, $ i =1, \dots, n$, which generate the Kahler isometries, can be explicitly written in the Plücker coordinates; however, for constructions we are interested only in the common level sets of the first $\widetilde \mu_1, \dots, \widetilde \mu_k$, and so, leaving aside the detailed analysis below, we will use the derived functions

By definition, the function $f_i$ has the following properties: its values vary from $0$ (for $\pi \subset H_i = \{ z_i = 0 \}$) to 1 (for $p_i \in \pi$); it does not admit other critical values. Adding the function $f_0$, we get the identity $\sum_{i =0}^n f_i = r+1$. Therefore, for the map

$$ \begin{equation*} F_{\mathrm{act}}= (f_1, \dots, f_n)\colon \operatorname{Gr} (r, n) \to \mathbb{R}^n = \mathbb{R}\langle x_1, \dots, x_n\rangle, \end{equation*} \notag $$

$$ \begin{equation*} P_{\operatorname{Gr}} = \biggl\{ (x_1, \dots, x_n) \in \mathbb{R}\langle x_1, \dots, x_n\rangle \biggm| 0 \leqslant x_i \leqslant 1,\, \sum_{i=1}^n x_i \leqslant r+1\biggr\}. \end{equation*} \notag $$

Our next strategy is to generalize the construction from [5]: for any $k$ from 1 to $n-r$ inclusive, we would like to construct a fibration into toric submanifolds with moment maps $f_1, \dots, f_k$, where, at the first step, we exclude some closed components from $\operatorname{Gr}(r, n)$. Such a fibration can be constructed as follows.

Let $p_1, \dots, p_k \in \mathbb{C} \mathbb{P}^n$ be the above stationary point for the toric action corresponding to the moment maps $f_1, \dots, f_k$. Since we will exploit projections from the projective span $\langle p_1, \dots, p_k \rangle$ to the projective span of its orthogonal complement $\langle p_0, p_{k+1}, \dots, p_n \rangle$ (in what follows, by $\langle \,{\cdots}\, \rangle$ we denote the projective spans of the sets of objects placed inside), and so, as the first step of our construction, we exclude from $\operatorname{Gr} (r, n)$ the points $[\pi]$ that correspond to the $r$-dimensional projective subspaces $\pi \subset \mathbb{C} \mathbb{P}^n$ satisfying the following properties: 1) $\pi \cap \langle p_1, \dots, p_k \rangle \neq \varnothing$; 2) $ \pi \subset H_i$, where $H_i = \{z_i = 0 \} \subset \mathbb{C} \mathbb{P}^n$ for $i =1, \dots, k$. Conditions 1) and 2) define in $\operatorname{Gr}(r, n)$ closed algebraic subvarieties; by cutting out these ones, we get an open algebraic variety $\operatorname{Gr}^0(r, n)$. We need the following result.

Lemma 1. All the excluded algebraic subvarieties are invariant under the Hamiltonian action of the moment maps $f_1, \dots, f_k$.

Indeed, case 2) corresponds directly to the zero critical value of $f_i$; this immediately implies the stationarity of $[\pi]$ with respect to the Hamiltonian action generated by $f_i$ itself. For each $\mu_j$, the toric action on the original $\mathbb{C} \mathbb{P}^n$ leaves invariant the basis hyperplane $H_i$, and so this action preserves the condition $\pi \subset H_i$, which implies the invariance of the subvariety

On the other hand, the projective subspace $\langle p_1, \dots, p_k \rangle \subset \mathbb{C} \mathbb{P}^n$ is also invariant with respect to the toric action, and the condition $\pi \cap \langle p_1, \dots, p_k \rangle \neq \varnothing$ is evidently preserved by the toric action. As a result, the subvariety

$$ \begin{equation*} A_k = \{[\pi] \in \operatorname{Gr}(r, n) \mid \pi \cap \langle p_1, \dots, p_k \rangle \neq \varnothing \} \subset \operatorname{Gr} (r, n) \end{equation*} \notag $$

The next step of our construction is to describe toric leaves — these being submanifolds of complex dimension $k$ in $\operatorname{Gr}^0(r, n)$ invariant with respect the Hamiltonian action of $f_1, \dots, f_k$.

Consider the projection of the $r$-dimensional projective subspace $\pi \subset \mathbb{C} \mathbb{P}^n$ corresponding to a point $[\pi] \in \operatorname{Gr}^0(r, n)$. This projection acts from the projective subspace $\langle p_1, \dots, p_k \rangle$ to the projective subspace $\langle p_0, p_{k+1}, \dots, p_n \rangle$; by the condition $\pi \cap \langle p_1, \dots, p_k \rangle = \varnothing$, and hence the result of projection is an $r$-dimensional projective subspace $\pi_0 \subset \langle p_0, p_{k+1}, \dots, p_n \rangle$. Consider the projective span

$$ \begin{equation*} L_{\pi} = \langle p_1, \dots, p_k, \pi \rangle = \langle p_1, \dots, p_k, \pi_0 \rangle \subset \mathbb{C} \mathbb{P}^n, \end{equation*} \notag $$

In the projective space $L_{\pi}$, with our subspace $\pi$ we associate the following set of data: we project our subspace $\pi$ from $\langle p_1, \dots, \widehat p_i, \dots, p_k \rangle$ to $\pi_0$ (here, $\widehat i$ denotes the exclusion of the point $p_i$ from the set). Since $\pi$ does not coincide with $\pi_0$ ($\pi_0$ is contained in any of $H_i$), the projection is $\alpha_i \subset \pi_0$, which is an $(r-1)$-dimensional projective subspace in $\pi_0$. By varying $i$ from $1$ to $k$, we get the set $(\alpha_1, \dots, \alpha_k)$ of hyperplanes in $\pi_0$.

In these terms, our subspace $\pi$ is distinguished by the following condition: for each $i = 1, \dots, k$, the intersection $\pi \cap \langle p_1, \dots, \widehat p_i, \dots, p_k, \alpha_i \rangle$ has projective dimension $r-1$. Indeed, the projective codimension of the span $\langle p_1, \dots, \widehat p_i, \dots, p_k, \alpha_i \rangle$ in $L_{\pi}$ is 2; therefore, the codimension of its intersection with $\pi$ is either 0 or 1. But the zero dimension would mean that $\pi \subset H_i$, which is excluded from the beginning.

Consider the open part $\operatorname{Gr}^0(r, L_{\pi}) = \operatorname{Gr}^0(r, n) \cap \operatorname{Gr}(r, L_{\pi})$ of the Grassmannian formed by $r$-dimensional projective subspaces in the projective space $L_{\pi}$. Consider a pencil $O_i$ of hyperplanes in $L_{\pi}$ that contain the subspace $\langle p_1, \dots, \widehat p_i, \dots, p_k, \alpha_i \rangle$. From the previous remark it follows that in the pencil $O_i$ there exists a unique hyperplane that contains $\pi$. By the assumption, $\pi$ does not intersect $\langle p_1, \dots, p_k \rangle$ and does not lie in $H_i$, and so from the pencil $O_i$ one can exclude the hyperplanes $\langle p_1, \dots, p_k, \alpha_k \rangle$ and $\langle p_1, \dots, \widehat p_i, \dots, p_k, \pi_0 \rangle$. Let $K_i$ be the hyperplane from the pencil $O_i$ that does contain the original subspace $\pi$.

Lemma 2. The set of hyperplanes $K_1, \dots, K_k$ from the pencils $O_1, \dots, O_k$ uniquely determines the subspace $\pi$.

Indeed, excluding from each $O_i$ the pair of hyperplanes $ \langle p_1, \dots, p_k, \alpha_k \rangle$ and $\langle p_1, \dots, \widehat p_i, \dots, p_k, \pi_0 \rangle$ we reduce the problem to the case where all possible $K_i$ are transversal, and so their intersection should have dimension $r$, but by the construction this intersection must contain our $\pi$, which implies Lemma 2.

So, we refine projection (4) as follows: $\alpha_i \subset \pi_0$ is the set of points in the dual projective space. Now the fibration

$$ \begin{equation} q\colon \operatorname{Gr}^0(r, n) \to \operatorname{tot} \bigl(\mathbb{P}(\tau^*) \times \dots \times \mathbb{P}(\tau^*) \to \operatorname{Gr} (r, n-k)\bigr) \end{equation} \tag{5} $$

At the same time, above we have described the fibres of the map $q$: the pencil $O_i$ without two elements is $\mathbb{C}^*$, and so the fibre is given by the direct product of $k$ copies of $\mathbb{C}^*$.

Indeed, for each $i = 1, \dots, k$, the corresponding pencil of hyperplanes $O_i$ is uniquely defined by its base set $\langle p_1, \dots, \widehat p_i, \dots, p_k, \alpha_i \rangle$. Therefore, it suffices to show that, for each $f_j$, the corresponding $U(1)$-action preserves this base set. But all the points from the set $\{ p_1, \dots, p_k \}$ are stationary under the Hamiltonain action of each $f_j$, and $\alpha_i \subset \langle p_0, p_{k+1}, \dots, p_n \rangle$ belongs to the critical set with zero critical value. Consequently, it is unchanged as well. Therefore, the pencil $O_i$ under the action of $T^k$ is mapped to itself. However, the elements of the pencil that correspond to the hyperplanes $\langle p_1, \dots, p_k, \alpha_k \rangle$ and $\langle p_1, \dots, \widehat p_i, \dots, p_k, \pi_0 \rangle$ are fixed in the pencil for this action. Now is clear that the intersections of elements of the corresponding open parts of $O_i$ should be invariant under this $T^k$-action. This completes the proof of Lemma 3.

The fibres of map $q$ are presented by open submanifolds $Y^0(\pi_0, \alpha_1, \dots, \alpha_k)$ parameterized by data sets formed by $r$-dimensional projective subspace $\pi_0 \subset \langle p_0, p_{k+1}, \dots, p_n \rangle$ together with the hyperplanes $(\alpha_1, \dots, \alpha_k)$ in this subspace. As was shown above, $Y^0(\pi_0, \alpha_1, \dots, \alpha_k)$ is isomorphic to $\mathbb{C}^* \times \dots \times \mathbb{C}^*$, and the torus $T^k$ spanned by the moment map acts freely on it on it. This implies that each fibre is a toric manifold (for properties of the toric action, see, for example, [7]). Therefore, we found toric leaves of the given $T^k$-action on $\operatorname{Gr}^0(r, n)$.

According to Liouville–Arnold theorem, the choice of common values $(c_1, \dots, c_k)$ for the moment maps $f_1, \dots, f_k$ cuts, in each fibre $Y^0(\pi_0, \alpha_1, \dots, \alpha_k)$, the corresponding smooth Lagrangian torus $T^k$. Therefore, we can describe, in terms of fibration (5), the common level set $N(c_1, \dots, c_k) \subset \operatorname{Gr}^0(r, n)$. Here, it is worth recalling that, for the construction of Lagrangian submanifolds in $\operatorname{Gr}(r, n)$ by Mironov’s method, we need to consider the whole Grassmannian. Therefore, this method can be exploited in terms of fibration (5) if it is possible to find value sets $(c_1, \dots, c_k)$ for the moment maps $f_1, \dots, f_k$ such that the common level set $N(c_1, \dots, c_k) \subset \operatorname{Gr}(r, n)$ should lie precisely in the open part $\operatorname{Gr}^0(r, n)$.

Lemma 4. Let $c_1, \dots, c_k$ satisfy s $0< c_i <1, \sum_{i=1}^k c_i < 1$. Then the common level set $N(c_1, \dots, c_k)$ is completely contained in the open part $\operatorname{Gr}^0(r, n)$.

Indeed, the components $B_i$ from Lemma 1 correspond to zero values of $f_i$ (they can be easily excluded). Now, let us show that the condition $\sum_{i=1}^k c_k < 1$ excludes the non-trivial intersection of $\pi$ and $\langle p_1, \dots, p_k \rangle$. Assume on the contrary that $\pi \cap \langle p_1, \dots, p_k \rangle$ is non-empty and contains as least one point $p$. Then the corresponding vector $v_p \in \mathbb{C}\langle e_1, \dots, e_k\rangle $ expands in vectors $e_i$. Hence

So, we have shown that the common level set of our functions $N(c_1, \dots, c_k)$ for $(c_1, \dots, c_k)$ defined in Lemma 4 is fibred into $k$-dimensional tori such that

$$ \begin{equation} q_{c_1, \dots, c_k}\colon N(c_1, \dots, c_k) \to \operatorname{tot} \bigl(\mathbb{P}(\tau^*) \times \dots \times \mathbb{P}(\tau^*) \to \operatorname{Gr} (r, n-k)\bigr), \end{equation} \tag{6} $$

Theorem 2. For the value set $c_1, \dots, c_k$ defined in Lemma 4, the result of symplectic reduction $\operatorname{Gr}(r, n)// T^k$ is the manifold $\operatorname{tot} (\mathbb{P}(\tau^*) \times \dots \times \mathbb{P}(\tau^*) \to \operatorname{Gr} (r, n-k))$.

Remark. It is worth pointing out that the symplectic form which appears on the reduced manifold does not in general coincide with the standard one; in addition, even for the direct summands (fibres over $\operatorname{Gr}(r, n-k)$, which are isomorphic to each other by construction), the restrictions of the reduced symplectic form can be different.¹

Now let us come back to our main aim of construction of Lagrangian submanifolds in $\operatorname{Gr} (r, n)$ via Mironov’s method. To this end, by definition, we need first to describe $S_{\mathbb{R}}(c_1, \dots, c_k) = N(c_1, \dots, c_k) \cap \operatorname{Gr}_{\mathbb{R}}(r, n)$, and then propagate this real submanifold by the toric action of our $T^k$. However, the same result can be reached by another method, where we avoid discussing smoothness properties of the constructed Lagrangian submanifold.

Lemma 5. The space $T^k(S_{\mathbb{R}}(c_1, \dots, c_k))$ for the values $c_1, \dots, c_k$ given in Lemma 4 is exactly the restriction of fibration (6) to the submanifold

$$ \begin{equation*} \operatorname{tot} \bigl(\mathbb{P}(\tau_{\mathbb{R}}^*) \times \dots \times \mathbb{P}(\tau^*_{\mathbb{R}}) \to \operatorname{Gr}_{\mathbb{R}} (r, n-k)\bigr) \subset \operatorname{tot} \bigl(\mathbb{P}(\tau^*) \times \dots \times \mathbb{P}(\tau^*) \to \operatorname{Gr} (r, n-k)\bigr), \end{equation*} \notag $$

which is the real part of the base of fibration (6).

To prove this result, we need to show the following: let $[\pi]$ be an arbitrary point of $T^k(S_{\mathbb{R}})$, then this point must lie in the fibre of $q_{c_1, \dots, c_k}$ over the uniquely determined point of the real part of the fibration base in (6).

By construction, fibration (6) inherits the anti-holomorphic conjugation: the moment maps $f_i$ are real functions, and hence the common level set $N(c_1, \dots, c_k)$ is invariant under the involution $\tau_{Gr}: \operatorname{Gr}(r, n) \to \operatorname{Gr}(r, n)$ induced by the standard involution $\tau\colon \mathbb{C} \mathbb{P}^n \to \mathbb{C} \mathbb{P}^n$; at the same time on the base $\operatorname{tot} (\mathbb{P}(\tau^*) \times \dots \times \mathbb{P}(\tau^*) \to \operatorname{Gr}(r, n-k))$ the action of $\tau_{Gr}$ is also compatible with the standard conjugation. This shows that $S_{\mathbb{R}}(c_1, \dots, c_k) \subset N(c_1, \dots, c_k)$ is exactly the real part of the common level set $N(c_1, \dots, c_k)$. Essentially, it remains to verify that the image $q_{c_1, \dots, c_k}(S_{\mathbb{R}}(c_1, \dots, c_k))$ is precisely $\operatorname{tot} (\mathbb{P}(\tau_{\mathbb{R}}^*) \times \dots \times \mathbb{P}(\tau^*_{\mathbb{R}}) \to \operatorname{Gr}_{\mathbb{R}} (r, n-k))$, that is, the real part of the base.

Consequently, we need to show that $\pi \in \operatorname{Gr}^0(r, n)$ is real if and only if the data set $(\pi_0, \alpha_1, \dots, \alpha_k)$ consists of real elements. But $\pi_0 \subset \langle p_0, p_{k+1}, \dots, p_n \rangle$ is given as the projection from a real projective subspace to a real projective subspace, and, therefore, $\pi$ and $\pi_0$ can be real only simultaneously. Since $\alpha_i$ are given by projections from real to real $\pi_0$, we get the same conclusion.

Therefore, the image $q_{c_1, \dots, c_k} (S_{\mathbb{R}}(c_1, \dots, c_k))$ is composed of the points that form the real part of the fibration base in (6). Since $T^k$ acts on the fibres of (6) in the standard way, the action generates all the fibres of (6) over the manifold $\operatorname{tot} (\mathbb{P}(\tau_{\mathbb{R}}^*) \times \dots \times \mathbb{P}(\tau^*_{\mathbb{R}}) \to \operatorname{Gr}_{\mathbb{R}} (r, n-k))$. This completes the proof of Lemma 5.

As a corollary to Theorem 1 and our last arguments, we get the following result: the Grassmannian $\operatorname{Gr}(r, n)$ equipped with the Plücker embedding with the corresponding Kahler structure admits $n-r+1$ different topological types realized by smooth Hamiltonian-minimal Lagrangian embeddings. Indeed, for each $k=0, \dots, n-r$, we choose values for the moment maps $f_1, \dots, f_k$ as in Lemma 4. Hence the corresponding submanifold $S_k$ is smooth qua restriction of a regular fibration to a smooth submanifold; on the other hand, this submanifold is realized by the Mironov construction as $T^k(S_{\mathbb{R}}(c_1, \dots, c_k))$. Therefore, by Theorem 1 this submanifold is Hamiltonian-minimal.

To complete the picture, we need to study the geometry of the constructed submanifolds $S_k$ for various $k$. To this end, we need to find the topological type of fibration $q$ from (5).

Note that this fibration can be written as $\operatorname{tot} \{ (P \times \dots \times P) \to \operatorname{Gr}(r, n-k) \}$, where $P = \operatorname{tot} (L \to X)$ is the total space of the corresponding $\mathbb{C}^*$-bundle over the projective space $X$, which is the projectivization of $\mathbb{P}(\tau^*)$. Indeed all the pencils $O_i$ are isomorphic to each other (this follows from the Kahler symmetries of $\operatorname{Gr}(r, n)$ generated by permutations of the homogenous coordinates $z_i \leftrightarrow z_j$ of the original projective space $\mathbb{C} \mathbb{P}^n$). Therefore, over each point of the base $\operatorname{Gr}(r, n-k)$ of the fibration (5) we have the bundle of the form $(L \times \dots \times L) \to (X \times \dots \times X)$, which is evidently isomorphic to the direct product $P \times \dots \times P$, where $P$ is the total space of the bundle $L \to X$.

So, we need to study the pencil $O_i$ and find how it depends on the choice of the hyperplane $\alpha_i \subset \pi_0$ so that all the information should be encoded by the subspace $\pi_0$, which defines the corresponding point of the base $\operatorname{Gr}(r, n-k)$ of fibration (5).

In the projective span $L_{\pi} = \langle p_1, \dots, p_k, \pi_0 \rangle \subset \mathbb{C} \mathbb{P}^n$, consider the space of hyperplanes passing trough the points $p_1, \dots, \widehat p_i, \dots, p_k$. We cut from this space the hyperplanes that contain either $\pi_0$ or $p_i$. The resulting subset is naturally fibred over the dual space $\pi_0^{\vee}$; moreover, this subset is precisely the total space $P$ of the corresponding bundle $L \to X$, which we have considered above. Indeed, if the hyperplane does not contain $\pi_0$, then it intersects $\pi_0$ along the corresponding $\alpha_i$; this is an element from the pencil $O_i$, as claimed.

Further, to describe the topological type of the total space $P$, we need to complete this $\mathbb{C}^*$-bundle over $\pi_0^{\vee}$ to the corresponding $\mathbb{C}$-bundle and find its topological type. The completion by hyperplanes in $L_{\pi}$ that contain $p_i$ does not alter the bundle structure over $\pi^{\vee}_0$, and, under this completion, each pencil $O_i$ defined by $\alpha_i \subset \pi_0$ gets exactly one element; on the other hand, a hyperplane in $L_{\pi}$ containing the point $p_ i$ together with the remaining $p_1, \dots, p_k$, may fail to contain the whole $\pi_0$, and, therefore, it is uniquely embedded in the corresponding pencil $O_i$. This implies that the completion of the $\mathbb{C}^*$-bundle $P$ to the $\mathbb{C}$-bundle $\widetilde P \to \pi^{\vee}_0$ is well defined. Changing to orthogonal objects, we attach the point $(\alpha_i)^{\perp} \in \pi_0$ of the hyperplane $\alpha_i \subset \pi_0$ to the hyperplanes $L_{\pi}$ from $O_i$. The corresponding fibration over $\pi_0$ is merely the attachment of a point $p \in \pi_0$ to the set $\langle p, p_i \rangle \setminus p_i \cong \mathbb{C}$, and this fibration is precisely the bundle ${\mathcal O}(1) \to \pi_0$. It remains to note that, in order to change back to the dual objects, we need to again conjugate this bundle. As a result, $\widetilde P$ is isomorphic to ${\mathcal O}(-1) \to \pi_0^{\vee}$. Recall that the Hopf bundle is the associated $U(1)$-bundle to $\widetilde P \to \pi_0^{\vee}$.

Let us now consider the “real” version (6) of fibration (5): in the fibre of $\widetilde P \to X$, we consider only the circle corresponding to our fixed set of values $(c_1, \dots, c_k)$, and from $\pi_0^{\vee}$ we take only the real part, which in the above analysis was corresponding to the fibre of $\mathbb{P}(\tau^*)$. It is not hard to see that as the real version of the total space of the bundle $\widetilde P$ we get precisely the Pushkar’ manifold $P_r \cong (S^1 \times S^r)/ \mathbb{Z}_2$, where $\mathbb{Z}_2$ acts diagonally by antipodal involutions on each direct summand. Indeed by construction, the result is given as the restriction of the Hopf bundle $S^{2r+1} \to \mathbb{C} \mathbb{P}^r$ to the real projective subspace $\mathbb{R} \mathbb{P}^r$; this is precisely the construction from [8].

Theorem 3. The Grassmann variety $\operatorname{Gr}(r, n)$ equipped with Kahler structure by Plücker embedding admits Hamiltonian-minimal smooth Lagrangian submanifolds $S_k$, $k = 0, \dots, n-r$, topologically isomorphic to bundles over $\operatorname{Gr}_{\mathbb{R}}(r, n-k)$ with a fibre given by the direct product of $k$ copies of the Pushkar’ manifold $P_r$.

Example. Putting $k = n-r$, we get as a corollary the existence of a Hamiltonian-minimal Lagrangian embedding of the direct product $(P_r \times \dots \times P_r) \subset \operatorname{Gr}(r, n)$ formed by $n-r$ copies of the Pushkar’ manifold.

On the other hand, our result is a direct generalization of Mironov’s result for $\mathbb{C} \mathbb{P}^n$: if we put $r=0$, then Theorem 3 gives a set of examples of Lagrangian submanifolds isomorphic to $T^k$-bundles over the corresponding bases $\mathbb{R} \mathbb{P}^{n-k}$. These constructions are precisely the new examples from [1] for $k =0, \dots, n$.

Corollary 1. The result of symplectic reduction $\operatorname{Gr}(r, n)// T^n$ can be naturally identified with $(\mathbb{C} \mathbb{P}^r \times \dots \times \mathbb{C} \mathbb{P}^r)// T^r$, where the torus $T^r$ acts diagonally on the direct product

Indeed, Theorem 2 for the case $k = n-r$ gives an isomorphism $\operatorname{Gr}(r, n)//T^{n-r} = \mathbb{C} \mathbb{P}^r \times \dots \times \mathbb{C} \mathbb{P}^r$, where the number of direct summands on the right is $n-r$. However, the toric action of the remaining $r$ moment maps is also reduced to the quotient manifold, and the further reduction by this action gives the desired result. Recall that the relation on the quotient manifolds, as given in the corollary, is well known as the Gel’fand–McPherson correspondence (see [9]). Earlier, this correspondence was found in the framework of the representation theory, and, in our construction, we derived it in a pure geometrical manner.

Our paper presents only one type of possible applications of the Mironov construction to an algebraic variety with natural toric action, but it is clear that the construction can also applied to flag varieties, moduli space of stable vector bundles over toric varieties, and to many other examples.

Citation: N. A. Tyurin, “Examples of Hamiltonian-minimal Lagrangian submanifolds in $\operatorname{Gr}(r, n)$”, Izv. Math., 89:2 (2025), 359–371

Citation in format AMSBIB

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\by N.~A.~Tyurin

\paper Examples of Hamiltonian-minimal Lagrangian submanifolds in~$\operatorname{Gr}(r, n)$

\jour Izv. Math.

\yr 2025

\vol 89

\issue 2

\pages 359--371

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Introduction

§ 1. Hamiltonian-minimal Lagrangian submanifolds

§ 2. The case of Grassmannian $\operatorname{Gr}(r, n)$

Bibliography