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In [1] Mironov proposed a new construction of the submanifolds in \( {𝔺}^{n} \) and \( {𝔺}{𝕇}^{n} \) satisfying the Lagrangian condition with respect to the Kähler forms induced by the constant metric and the Fubini–Study metric respectively. He was motivated by the problem of constructing minimal and Hamiltonian minimal Lagrangian immersions, i.e., Lagrangian subvarieties admitting self-intersections. Therefore, below we discuss Lagrangian cycles. However, the problem of constructing new Lagrangian cycles in algebraic manifolds is itself important both from the viewpoint of symplectic geometry and from the viewpoint of geometric quantization in mathematical physics. Mironov’s construction can be extended verbatim to a rather large class of algebraic manifolds. The necessary conditions for an algebraic manifold \( X \) of complex dimension \( n \) are the presence of an action of the real torus \( {𝕋}^{k} \) by Kähler isometries and the existence of an antiholomorphic involution such that the real part \( X_{{𝕉}} \) is a smooth submanifold of real dimension \( n \) transversal to the toric action. In this case, as [2] shows, a family of Lagrangian cycles can be constructed, which we call Mironov cycles.

This article deals with the first examples arising when we consider complex Grassmannians. Namely, as we will show, the algebraic variety \( \operatorname{Gr}(k,n+1) \), endowed with the Kähler form under the Plücker embedding into the projective space, enjoys both the natural toric action and the natural real structure. Applying Mironov’s construction in this situation, we obtain Lagrangian submanifolds, i.e., Mironov cycles. In this article we consider the simplest case: The homogeneity degree of the cycle equals 1, but there are interesting results already in this case. Subsequently we hope to describe all types of Mironov cycles in complex Grassmannians.

1. Generalization of Mironov’s Construction for Algebraic Manifolds

Suppose that \( (X,\omega) \) is a simply-connected compact smooth symplectic manifold of real dimension \( 2n \). Assume that our symplectic manifold admits an incomplete toric action, i.e., a tuple of smooth functions \( f_{1},\dots,f_{k} \), where \( k<n \), commuting with respect to the standard Poisson bracket and inducing via its Hamiltonian action some action of the \( k \)-dimensional torus \( {𝕋}^{k} \) on \( M \); furthermore, for this action the tuple \( (f_{1},\dots,f_{k}) \) is the tuple of moment maps. It is known that in the completely integrable case (i.e., when \( k=n \)) there are Lagrangian submanifolds composed of joint level sets of the moment maps: Liouville tori. In our case the classical theorem is inapplicable, but we would still like to learn how to construct Lagrangian submanifolds. Consider a generic tuple of values \( c_{1},\dots,c_{k} \) of the moment maps such that the joint level set \( N(c_{1},\dots,c_{k})=\{f_{i}=c_{i},i=1,\dots,k\} \) is disjoint from the determinant locus \( \Delta(f_{1},\dots,f_{k})=\{X_{f_{1}}\wedge\dots\wedge X_{f_{k}}=0\} \), where \( X_{f_{i}} \) is the Hamiltonian vector field of the function \( f_{i} \), and suppose that there is an isotropic \( (n-k) \)-dimensional submanifold \( S_{0}\subset N(c_{1},\dots,c_{k}) \) transversal to the toric action at each of its points. This condition means that for every point \( p\in S_{0}\subset N(c_{1},\dots,c_{k}) \) the tangent space \( T_{p}S_{0} \) and the linear span \( \langle X_{f_{1}}(p),\dots,X_{f_{k}}(p)\rangle \) of the Hamiltonian vector fields of the moment maps are transversal in the ambient space \( T_{p}N(c_{1},\dots,c_{k}) \).

Then the action of the torus on the submanifold \( S_{0} \) generates a real \( n \)-dimensional cycle \( {𝕋}^{k}(S_{0}) \), which is a Lagrangian immersion. Depending on the particular situation, it can turn out smooth or have self-intersections; moreover, in some cases we can also consider the special values of the moment map \( f_{i} \), for instance the critical ones, and obtain smooth Lagrangian submanifolds. The Lagrangian property of the cycles \( {𝕋}^{k}(S_{0}) \) at smooth points follows from a simple observation: the linear span \( \langle T_{p}S_{0},X_{f_{1}}(p),\dots,X_{f_{k}}(p)\rangle \) is a Lagrangian subspace in \( T_{p}M \) for each point \( p\in S_{0} \), while the Lagrangian condition is preserved by the Hamiltonian action of the torus \( {𝕋}^{k} \); thus, the same remains valid for all points in the orbit \( {𝕋}^{k}(p)\in{𝕋}^{k}(S_{0}) \). The details and the simplest examples are presented in [2].

Thus, in the case of an incomplete tuple \( (f_{1},\dots,f_{k}) \) of moment maps on a \( 2n \)-dimensional symplectic manifold \( (M,\omega) \) it is necessary to learn how to find a suitable isotropic submanifold \( S_{0} \) in the joint level set \( N(c_{1},\dots,c_{k})\subset M \). Mironov’s construction [1] suggests a possible path in the case that the real manifold \( (M,\omega) \) is subordinate to a complex algebraic manifold.

Each compact complex algebraic manifold \( X \), understood from the real viewpoint as a pair \( (X,I) \), where \( I \) is a suitable complex structure, can be naturally considered as a real symplectic manifold. By definition (see [3]) this \( X \) has a very ample line bundle \( L\to X \) such that the associated complete linear system \( |L| \) induces an embedding \( \phi_{L}:X\hookrightarrow{𝔺}{𝕇}^{N} \) into the corresponding projective space, and the pullback \( \omega_{L}=\phi_{L}^{*}\Omega_{FS} \) of the standard Kähler form \( \Omega_{FS} \) of the Fubini–Study metric is a symplectic (Kähler) form \( \omega_{L} \) on \( X \) which agrees with the complex structure. This form is obviously not unique, but the cohomology class \( [\omega_{L}]=c_{1}(L)\in H^{2}(X,{𝕑}) \) is fixed by the choice of \( L \) and the Lagrangian geometry of the pair \( (X,\omega_{L}) \) depends only on this class; i.e., the realization of the middle homology classes by Lagrangian submanifolds, the realization of \( n \)-dimensional topological types by Lagrangian submanifolds, classification problems, and so on. Note meanwhile that distinct ample bundles on \( X \) produce a priori distinct Lagrangian geometries marked by cohomology classes.

Suppose that a compact algebraic manifold \( (X,I,\omega_{L}) \) endowed with a symplectic form \( \omega_{L} \) admits a toric action by Kähler isometries, i.e., the Hamiltonian action of moment maps \( f_{1},\dots,f_{k} \) simultaneously preserves the complex structure of \( I \). This situation can be illustrated as follows. Fix a system of homogeneous coordinates \( [z_{0}:\dots:z_{N}] \) on the ambient projective space \( {𝔺}{𝕇}^{N} \) agreeing with the Kähler form \( \Omega_{FS} \) fixed above. Consider a nondegenerate integer matrix \( \Lambda=(\lambda_{ij}) \), for \( 0\leq i,j\leq N \), whose zeroth row \( (\lambda_{0j})=(1,\dots,1) \) consists of 1s, and consider the corresponding tuple \( N \) of the standard moment maps of the form

$$ F_{i}=\frac{\sum\limits_{j=0}^{N}\lambda_{ij}|z_{j}|^{2}}{\sum\limits_{j=0}^{N}|z_{j}|^{2}},\quad i=1,\dots,N,\ \lambda_{ij}\in{𝕑}. $$

Assume in addition that the matrix \( \Lambda \) is such that \( k \) moment maps \( F_{i} \) (without loss of generality we may assume that precisely the first \( k \)’s are the mappings \( F_{1},\dots,F_{k} \)) preserve the Hamiltonian action of the image \( \phi_{L}(X)\subset{𝔺}{𝕇}^{N} \). Then obviously the pullbacks \( \phi_{L}^{*}(F_{i}) \) induce the corresponding \( {𝕋}^{k} \)-action on \( (X,I,\omega_{L}) \) by Kähler isometries.

It turns out that for some algebraic manifolds the choice of an isotropic submanifold in the joint level set \( N(c_{1},\dots,c_{k}) \) can be made almost automatically. To this end, suppose that our algebraic manifold \( X \) carries one more structure, namely, an antiholomorphic involution \( \sigma:X\to X \). Pass again to the coordinates \( [z_{0}:\dots:z_{N}] \) and consider the mapping

$$ \bar{\sigma}:[z_{0}:\dots:z_{N}]\mapsto[\bar{z}_{0}:\dots:\bar{z}_{N}]. $$

Assume that \( \phi_{L}(X) \) is a real submanifold with respect to the involution \( \bar{\sigma} \), i.e., \( \bar{\sigma}(\phi_{L}(X))=\phi_{L}(X) \). It is not difficult to see that then the pullback \( \phi^{*}_{L}\bar{\sigma} \) is an antiholomorphic involution, and every prescribed \( \sigma \) is realized in exactly this form. Moreover, suppose that the real part \( X_{{𝕉}}=\{x\in X\mid\sigma(x)=x\}\subset X \) has the maximal possible real dimension \( n \).

It is not difficult to see that \( X_{{𝕉}}\subset X \) is Lagrangian with respect to the symplectic form \( \omega_{L} \). Consequently, for every tuple of values \( c_{1},\dots,c_{k} \) the intersection \( S_{{𝕉}}(c_{1},\dots,c_{k})=X_{{𝕉}}\cap N(c_{1},\dots,c_{k}) \) is an isotropic submanifold; moreover, the components \( S_{{𝕉}}(c_{1},\dots,c_{k}) \) are smooth and transversal to the action of the torus \( {𝕋}^{k} \) for generic values. Therefore, we can apply the above construction and obtain the Lagrangian cycles \( {𝕋}^{k}(S_{{𝕉}}(c_{1},\dots,c_{k})) \) in \( (X,I,\omega_{L}) \); for details, see [2].

Since the presented construction of Lagrangian cycles in an algebraic manifold is a direct generalization of the construction of minimal and Hamiltonian minimal Lagrangian immersions into \( {𝔺}^{n} \) and \( {𝔺}{𝕇}^{n} \) proposed in [1], we refer to \( {𝕋}^{k}(S_{{𝕉}}(c_{1},\dots,c_{k}))\subset(X,\omega_{L}) \) as Mironov cycles, or Mironov submanifolds in the smooth case.

2. Grassmann Manifolds

A toric action, complete or incomplete, on an algebraic variety arises in the case when the variety is built over a toric variety. For instance, the standard toric action on the projective space \( {𝔺}{𝕇}^{n} \) induces a toric action on the set of all projective subspaces of a fixed dimension. Thus, the Grassmann manifold \( \operatorname{Gr}_{{𝔺}}(k,n+1) \), consisting of all \( k \)-dimensional subspaces of \( {𝔺}^{n+1} \), carries the natural action of the \( n \)-dimensional torus.

The Grassmann manifold \( \operatorname{Gr}_{{𝔺}}(k,n+1) \) embeds into the projective space \( {𝕇}(\wedge^{k}{𝔺}^{n+1}) \) by the Plücker embedding (see [3]) and we can explicitly extract those moment maps in the latter which preserve the Grassmannian.

In the original space \( {𝔺}^{n+1} \), fix coordinates \( (Z_{0},\dots,Z_{n}) \) and the standard hermitian structure in this coordinate system. This fixes the corresponding Kähler structure on the projective space \( {𝔺}{𝕇}^{n} \); denote the corresponding homogeneous coordinates by \( [z_{0}:\dots:z_{n}] \). The standard toric action is induced by the moment maps

$$ \mu_{i}=\frac{|z_{i}|^{2}}{\sum\limits_{j=0}^{n}|z_{j}|^{2}},\quad i=1,\dots,n, $$

and this action obviously carries each \( (k-1) \)-dimensional projective subspace \( l\subset{𝔺}{𝕇}^{n} \) into another \( (k-1) \)-dimensional projective space. This action lifts explicitly to the ambient projective space \( {𝕇}(\wedge^{k}{𝔺}^{n+1}) \) because this space is endowed with distinguished Plücker coordinates \( w_{i_{1},\dots,i_{k}} \) (see [3]). The induced moment maps \( F(\mu_{i}) \) in these coordinates are of the form

$$ F(\mu_{i})=\frac{\sum\limits_{(i_{1},\dots,i_{k})}\delta(i,i_{1},\dots,i_{k})|w_{i_{1},\dots,i_{k}}|^{2}}{\sum\limits_{(i_{1},\dots,i_{k})}|w_{i_{1},\dots,i_{k}}|^{2}}, $$

where the symbol \( \delta(i,i_{1},\dots,i_{k}) \) equals 1 if \( i=i_{j} \) for some \( j \) and 0 otherwise. Indeed, the moment map \( \mu_{i} \) via its Hamiltonian action on \( {𝔺}{𝕇}^{n} \) induces the one-parameter subgroup of transformations of the form \( [z_{0}:\dots:e^{it}z_{i}:\dots:z_{n}] \) in the original homogeneous coordinates, where \( t \) is the parameter of the one-parameter subgroup. By the definition of Plücker coordinates the induced action is nontrivial only on the coordinate \( w_{i_{1},\dots,i_{k}} \) containing \( i \) among its indices.

The geometric meaning of the induced moment maps \( F(\mu_{i}) \) is very simple: the value at \( [L]\in\operatorname{Gr}_{{𝔺}}(k,n+1) \) representing the subspace \( L\subset{𝔺}^{n+1} \) equals

$$ F(\mu_{i})([L])=\max_{v\in L}\frac{|\langle v_{i},v\rangle|}{|v|}, $$

where \( v_{i} \) is the \( i \)th basis vector in \( {𝔺}^{n+1} \). In other words, this value corresponds to the distance from the point \( [0:\dots:1:\dots 0] \), where 1 appears in the slot \( i \), to the corresponding projective subspace \( l={𝕇}(L) \) in \( {𝔺}{𝕇}^{n} \) with respect to the standard Fubini–Study metric; i.e., essentially, \( F(\mu_{i}) \) measures the angle between \( L \) and \( v_{i} \).

Hence, it is clear that \( F(\mu_{i}) \) has precisely two critical values: 0 and 1. The first critical value corresponds to the case that \( L \) lies in the orthogonal complement \( \langle v_{i}\rangle^{\perp}\subset{𝔺}^{n+1} \); and the second, to the case \( v_{i}\in L \). All other values \( 0<c_{i}<1 \) are not critical: The Hamiltonian vector field \( X_{F(\mu_{i})} \) does not vanish on the level set \( N(c_{i})=\{F(\mu_{i})=c_{i}\}\subset\operatorname{Gr}_{{𝔺}}(k,n+1) \).

On the other hand, it is known that the real part \( \operatorname{Gr}_{{𝕉}}(k,n+1)\subset\operatorname{Gr}_{{𝔺}}(k,n+1) \) is well defined by construction, has the right dimension, and is transversal to the toric action induced by the tuple of moment maps \( F(\mu_{i}) \). Indeed, the original complex space \( {𝔺}^{n+1} \) with the fixed coordinates has the real part \( {𝕉}^{n+1} \) specified by the conditions \( Z_{i}\in{𝕉} \). Furthermore, in the ambient projective space \( {𝕇}(\wedge^{k}{𝔺}^{n+1}) \) this naturally selects the real part \( {𝕇}(\wedge^{k}{𝕉}^{n+1}) \) transversal to the toric action of the moment maps \( F(\mu_{i}) \), and the transversality remains under restriction to the real submanifold \( \operatorname{Gr}_{{𝕉}}(k,n+1) \). From the projective points of view, for every real projective subspace \( l\subset{𝕉}{𝕇}^{n}\subset{𝔺}{𝕇}^{n} \) the flow \( \phi^{t}_{X_{\mu_{i}}} \) induced by the Hamiltonian vector field \( X_{\mu_{i}} \) of the function \( \mu_{i} \) either acts trivially on \( l \) or translates \( l \) outside \( {𝕉}{𝕇}^{n} \). The trivial action corresponds to the case that \( F(\mu_{i})([l]) \) equals 0 or 1. We exclude this case from further consideration. Otherwise, the projective subspace \( l \) must contain at least one real point with nonzero \( i \)th coordinate; the corresponding flow multiplies this coordinate by \( e^{it} \) while keeping the remaining ones intact. Hence, it is clear that this point must leave \( {𝕉}{𝕇}^{n} \), even though the point becomes real again when \( t \) reaches \( \pi \).

The real Grassmannian \( \operatorname{Gr}_{{𝕉}}(k,n+1) \) has the right real dimension, coinciding with the complex dimension of \( \operatorname{Gr}_{{𝔺}}(k,n+1) \). Therefore, it is clear that Mironov’s construction can be correctly applied in this case.

3. Mironov Lagrangian Submanifolds

Each action of the \( n \)-dimensional torus can always be restricted to an action of some subtorus \( {𝕋}^{k} \), for \( k\leq n \), which corresponds to the choice of some rank \( k \) subsystem of moment maps; Mironov’s construction can be applied to any subsystem. To distinguish the cases, we introduce the concept of homogeneity degree for Mironov cycles: This number equals the rank of the subsystem of moment maps \( \tilde{f}_{1},\dots,\tilde{f}_{k} \) participating in the definition of \( {𝕋}^{k}(S_{{𝕉}}(c_{1},\dots,c_{k})) \). For instance, if we choose the sole mapping \( \tilde{f}_{1} \) then the corresponding Mironov cycle \( {𝕋}^{1}(S_{{𝕉}}(c_{1}) \) has homogeneity degree 1.

Below we will give an example of constructing a Mironov Lagrangian cycle of homogeneity degree 1 in the complex Grassmannian \( \operatorname{Gr}_{{𝔺}}(k,n+1) \). As the moment map we consider \( F(\mu_{1}) \); obviously, the analogous construction works for each \( F(\mu_{i}) \).

The function \( F(\mu_{1}) \) agrees with the following decomposition used to define the topology of Grassmann manifolds. Namely, for a fixed basis vector \( v_{1}\in{𝕉}^{n+1}\subset{𝔺}^{n+1} \) the Grassmannian \( \operatorname{Gr}_{{𝕂}}(k,n+1) \), for both \( {𝕂}={𝕉} \) and \( {𝕂}={𝔺} \), is decomposed as

$$ \operatorname{Gr}_{{𝕂}}(k,n+1)=\operatorname{Gr}_{{𝕂}}(k-1,n+1)\cup\operatorname{tot}(E_{{𝕂}}\to\operatorname{Gr}_{{𝕂}}(k,n)), $$
(1)

where \( E_{{𝕂}} \) is a rank \( k \) vector \( {𝕂} \)-bundle, while \( \operatorname{tot} \) stands for the total space of the bundle \( E_{{𝕂}} \); see [3]. The first component consists of the subspaces \( L \) containing the vector \( v_{1} \), and so it coincides with \( F(\mu_{1})^{-1}(1) \). The second component consists of the subspaces avoiding \( v_{1} \). Clearly, we can express it as a vector bundle over \( \operatorname{Gr}_{{𝕂}}(k,n) \), the Grassmannian of \( k \)-dimensional subspaces in \( {𝕂}\langle v_{1}\rangle^{\perp} \) whose zero section is precisely \( F(\mu_{1})^{-1}(0) \).

Fix an arbitrary noncritical value \( c_{1}\in(0;1) \). The first step in the construction is to describe the intersection \( X_{{𝕉}}\cap N(c_{1})=\operatorname{Gr}_{{𝕉}}(n,k+1)\cap\{F(\mu_{1})=c_{1}\} \) denoted by \( S_{{𝕉}}(c_{1}) \). Below we use the version of (1) with \( {𝕂}={𝕉} \).

Since the choice of \( c_{1} \) excludes the first component of (1), we can recover each \( [L]\in S_{{𝕉}}(c_{1}) \) as follows.

Recall the description of the second component of (1): the base of the bundle \( E_{{𝕉}}\to\operatorname{Gr}_{{𝕉}}(k,n) \) is the variety of all \( k \)-dimensional subspaces in the orthogonal complement \( \langle v_{1}\rangle^{\perp}\subset{𝕉}^{n+1} \). The fiber \( E_{{𝕉}}|_{L_{0}} \) over each subspace \( L_{0}\subset\langle v_{1}\rangle^{\perp} \) consists of the \( k \)-dimensional subspaces of the direct sum \( {𝕉}\langle v_{1}\rangle\oplus L_{0} \) avoiding the vector \( v_{1} \). The set of all these subspaces is parametrized by \( {𝕉}{𝕇}^{k} \) and, excluding the subspace containing \( v_{1} \), we obtain exactly a \( k \)-dimensional real vector space, the fiber of the bundle \( E_{{𝕉}} \).

The subspace \( L\in{𝕉}^{n+1} \) representing an element of \( E_{{𝕉}}|_{[L_{0}]} \) is almost uniquely determined by two pieces of data: the intersection \( L\cap L_{0}\subset{𝕉}\langle v_{1}\rangle\oplus L_{0} \) and the angle under which the vector \( v_{1} \) is seen from \( L \). If the angle is zero or right then this correspondence is unique; if the angle differs from zero or right then we have precisely two possibilities for choosing \( L \) for a fixed intersection with \( L_{0} \). Since \( c_{1}\neq 0,1 \), in the case of interest to us the angle is neither zero nor right. For a fixed intersection \( L\cap L_{0}=M\subset L_{0} \) a suitable \( L \) is reconstructed as follows: The normal vector to \( L \) in \( {𝕉}\langle v_{1}\rangle\oplus L_{0} \) is orthogonal to \( M \) and has a fixed angle with \( v_{1} \), and so there are precisely two possibilities open. All possible intersections \( M\subset L_{0} \) constitute the projectivization of the dual space to \( L_{0} \). Thus, \( S_{{𝕉}}(c_{1})\cap E_{{𝕉}}|_{[L_{0}]} \) is isomorphic to the two-sheeted cover \( S(L_{0}^{*})\to{𝕇}(L_{0}^{*}) \), i.e., a \( (k-1) \)-dimensional sphere.

Recall that the Grassmann manifold \( \operatorname{Gr}_{{𝕉}}(k,n) \) admits the natural tautological bundle \( \tau \): Its fiber \( \tau|_{[L]} \) is \( L \) itself; see [3]. The following description of \( S_{{𝕉}}(c_{1}) \) gives the globalization of the above picture: This set is a bundle over \( \operatorname{Gr}_{{𝕉}}(k,n) \) with fiber isomorphic to the spherization of the fiber of \( \tau^{*} \); i.e.,

$$ S_{{𝕉}}(c_{1})=\operatorname{tot}(S^{k-1}(\tau^{*})\to\operatorname{Gr}_{{𝕉}}(k,n))\subset\operatorname{Gr}_{{𝕉}}(k,n+1). $$

The second step of the construction is to include the toric action induced by the Hamiltonian vector field \( X_{F(\mu_{1})} \). First of all, note that this toric action, applied to \( S_{{𝕉}}(c_{1}) \), preserves the structure of bundle over \( \operatorname{Gr}_{{𝕉}}(k,n) \). Indeed, the decomposition (1) into components over \( {𝕉} \) and over \( {𝔺} \) agrees with the complexification; hence, the toric action preserves the fibers \( E_{{𝔺}}\to\operatorname{Gr}_{{𝔺}}(k,n) \), and so

$$ \phi^{t}_{X(\mu_{i})}(L)\in E_{{𝔺}}|_{[L_{0}]}, $$

where \( [L_{0}]\in Gr_{{𝕉}}(k,n)\subset\operatorname{Gr}_{{𝔺}}(k,n) \). We infer that the Mironov cycle \( {𝕋}^{1}(S_{{𝕉}}(c_{1})) \) is a bundle over \( \operatorname{Gr}_{{𝕉}}(k,n) \). Let us describe its fiber.

Over an arbitrary point \( [L_{0}]\in\operatorname{Gr}_{{𝕉}}(k,n) \) we have the sphere \( S^{k-1} \) in the fiber \( {𝕉}^{k} \) of the bundle \( \tau_{{𝕉}}^{*} \). By construction, the antipodal points of this sphere correspond to the pairs of real subspaces \( L_{1},L_{2}\subset{𝕉}\langle v_{1}\rangle\oplus L_{0} \). The toric action \( F(\mu_{1}) \) multiplies \( v_{1} \) by \( e^{it} \), which implies that the flow \( \phi^{t}_{X_{F(\mu_{1})}} \) switches \( L_{1} \) and \( L_{2} \) at time \( t=\pi \), while for all other \( t\neq\pi m \) with \( m\in{𝕑} \) the circular trajectory does not cross the sphere \( S^{k-1} \). Thus, the fiber is \( (S^{1}\times S^{k-1})/{𝕑}_{2} \), the quotient of the direct product of spheres by the simultaneous action of antipodal involutions.

The manifolds of this type were studied by Pushkar [4]. He showed that this manifold is isomorphic to \( S^{1}\times S^{k-1} \) for \( k \) even and is a nonorientable manifold for \( k \) odd; see Proposition 1 of [4]. In any case this is a topologically nontrivial \( U(1) \)-bundle over \( {𝕉}{𝕇}^{k-1} \) called a generalized Klein bottle; see [1]. Note that Pushkar’s construction is a particular case of Mironov’s construction [1].

Let us globalize the picture over the whole \( \operatorname{Gr}_{{𝕉}}(k,n) \): The toric action induced by \( F(\mu_{1}) \) is tangent only to \( {𝕉}\langle v_{1}\rangle \). Therefore, it corresponds to the trivial bundle \( \underline{S^{1}}\to\operatorname{Gr}_{{𝕉}}(k,n) \). The following theorem contains the general result.

Theorem

The Mironov Lagrangian cycle \( {𝕋}^{1}(S_{{𝕉}}(c_{1}))\subset\operatorname{Gr}_{{𝔺}}(k,n+1) \) induced by the moment map \( F(\mu_{1}) \) for each value \( c_{1}\in(0;1) \) is a smooth manifold isomorphic to

$$ {𝕋}^{1}(S_{{𝕉}}(c_{1}))=\operatorname{tot}(\underline{S}^{1}\times S^{k-1}(\tau^{*}))/{𝕑}_{2}\to\operatorname{Gr}_{{𝕉}}(k,n)), $$

where \( \tau^{*} \) is the bundle dual to the tautological bundle.

Proof

Indeed, as we showed above, we obtain the required Mironov cycle as follows. Take the direct sum \( \underline{S}^{1}\times S^{k-1}(\tau^{*})\to\operatorname{Gr}_{{𝕉}}(k,n)) \) of the trivial \( S^{1} \)-bundle and the spherization of \( \tau^{*} \); consider the fiberwise diagonal action of \( {𝕑}_{2} \) defined as the simultaneous action of the standard antipodal involutions on both spheres; take the quotient by this action to obtain the answer. The result of all these operations is obviously a smooth manifold.  ☐

4. Examples

Let us exhibit some examples of Mironov Lagrangian cycles that are obtained from our construction.

Consider the simplest case \( k=1 \), i.e., the Grassmann manifold \( \operatorname{Gr}_{{𝔺}}(k,n+1) \) is \( {𝔺}{𝕇}^{n} \). Then our construction yields the following Mironov cycle. The spherization of the dual to the tautological bundle yields a bundle whose fiber is a pair of points (zero-dimensional sphere) over \( {𝕉}{𝕇}^{n-1} \), which globally is a sphere \( S^{n-1} \); since the \( S^{1} \)-bundle is topologically trivial, the pointwise factorization \( (S^{1}\times S^{0})/{𝕑}_{2} \) of the direct product of the fibers and further globalization over the base yield the same result as the factorization \( (S^{1}\times S^{n-1})/{𝕑}_{2} \), where \( S^{n-1} \) is the total space of the bundle \( S^{0}\to{𝕉}{𝕇}^{n-1} \). We obtain a generalized Klein bottle, as in [1].

The case \( k=2 \) is most interesting in light of the conjecture of [2]. For the Grassmann manifold \( \operatorname{Gr}_{{𝔺}}(2,n+1) \) our construction yields a smooth Lagrangian submanifold of the following topological type. It is the total space of a topologically nontrivial \( {𝕋}^{2} \)-bundle over \( \operatorname{Gr}_{{𝕉}}(2,n) \). Indeed, the tautological bundle in this case has rank 2; according to Pushkar’s result the corresponding quotient is a two-dimensional torus and the bundle \( (\underline{S}^{1}\times S^{1}(\tau^{*}))\backslash{𝕑}_{2}\to\operatorname{Gr}_{{𝕉}}(2,n) \) is associated to the tautological one.

For the Grassmannian \( \operatorname{Gr}_{{𝔺}}(2,3)\simeq{𝔺}{𝕇}^{2} \) this yields a standard Liouville torus, the \( {𝕋}^{2} \)-bundle over a point; for the Grassmannian \( \operatorname{Gr}_{{𝔺}}(2,4) \) our construction yields a smooth Lagrangian submanifold fibered over the projective space \( \operatorname{Gr}_{{𝕉}}(2,3)={𝕉}{𝕇}^{2} \) with fiber \( {𝕋}^{2} \). This manifold can be more explicitly described as follows: Consider \( {𝕉}^{3} \) with the standard Euclidean metric which determines the unit sphere \( S^{2}_{e}\subset{𝕉}^{3} \). Each point \( p\in S^{2}_{e} \) of the sphere determines the polar plane \( \pi(p)\subset{𝕉}^{3} \); furthermore, the antipodal points determine the same plane, and so \( \operatorname{Gr}_{{𝕉}}(2,3) \) can be identified with the quotient of \( S^{2}_{e} \) by the antipodal involution. In this situation the dual bundle to the tautological bundle \( \tau^{*} \) is explicitly realized as the induced quotient of the cotangent bundle to \( S^{2}_{e} \) modulo the antipodal involution, and the same holds for the spherizations of these bundles. Then it is not difficult to express the submanifold of our theorem as a bundle over \( S^{2}_{e} \) with fiber \( {𝕇}\bigl{(}T^{*}_{p}S^{2}_{e}\bigr{)} \). This transition agrees with “turning on” the toric action: The projectivization of the fiber \( T^{*}_{p}S^{2}_{e} \) leaves one point from the circle \( S^{1}\bigl{(}T^{*}_{p}S^{2}_{e}\bigr{)} \), which precisely corresponds to the factorization with respect to the real points of the phase rotation induced by \( X_{F(\mu_{1})} \). Hence, we infer that the corresponding Mironov cycle in \( \operatorname{Gr}_{{𝔺}}(2,4) \) is isomorphic to \( S^{1}\times{𝕇}(T^{*}S^{2}) \).

In the general case \( \operatorname{Gr}_{{𝔺}}(2,n+1) \) we can use the previous example as follows. Since the Grassmannians \( \operatorname{Gr}_{{𝕂}}(2,n+1) \) and \( \operatorname{Gr}_{{𝕂}}(n-1,n+1) \) are isomorphic, construct a Mironov Lagrangian cycle in the latter and then push it to the first one. Our theorem shows that the corresponding Mironov cycle in \( \operatorname{Gr}_{{𝔺}}(n-1,n+1) \) is a bundle over \( \operatorname{Gr}_{{𝕉}}(n-1,n) \), i.e., the projective space \( {𝕉}{𝕇}^{n-1} \). Consider once again the vector space \( {𝕉}^{n} \) with the standard Euclidean structure and the unit sphere \( S^{n-1}_{e} \) in \( {𝕉}^{n} \). The spherization of the bundle \( \tau^{*} \) dual to the tautological bundle \( \tau \) can be represented by the pairs \( (p,s) \), where \( p \) is a point on the sphere \( S^{n-1}_{e} \) modulo the antipodal involution and \( s \) is a point on the unit sphere \( S^{n-2}_{p}\in\bar{\pi}(p) \), where \( \bar{\pi}(p)\subset{𝕉}^{n} \) is the \( n-1 \)-dimensional hyperplane passing through \( p \) and tangent to the sphere \( S^{n-1}_{e} \). It is obvious that instead of the involution on \( S^{n-1}_{e} \) we can consider the conjugation of points on the spheres \( S^{n-2}_{p} \). This conjugation agrees with the toric action, whence we obtain a Mironov Lagrangian cycle isomorphic to \( S^{1}\times{𝕇}(T^{*}S^{n-1}) \).

Note that we actually considered rather simple examples of Mironov cycles of homogeneity degree 1. As the selected moment map we chose \( F(\mu_{1}) \). This choice is far from general because every linear combination

$$ \sum\limits_{j=1}^{n}\alpha_{j}F(\mu_{j}) $$

is a moment map of the general form, which is easy to see because for \( \operatorname{Gr}_{{𝔺}}(2,4) \) we gave in [2] examples of some other topological types of Mironov Lagrangian cycles that differ from those in this article. Thus, it would be natural to expect a great variety of new examples of Lagrangian submanifolds obtained from the proposed generalization of Mironov’s construction.