Rigid subsets of symplectic manifolds
Abstract
We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P.Albers and P.BiranO.Cornea), as well as certain, possibly singular, sets defined in terms of Poissoncommutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semisimplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floertheoretical machinery of partial symplectic quasistates.
Contents
 1 Introduction and main results
 2 Detecting stable displaceability
 3 Preliminaries on Hamiltonian Floer theory
 4 Basic properties of (super)heavy sets
 5 Products of (super)heavy sets
 6 Stable nondisplaceability of heavy sets
 7 Analyzing stable stems
 8 Monotone Lagrangian submanifolds
 9 Rigidity of special fibers of Hamiltonian actions
1 Introduction and main results
1.1 Many facets of displaceability
A wellstudied and easy to visualize rigidity property of subsets of a symplectic manifold is the rigidity of intersections: a subset cannot be displaced from the closure of a subset by a compactly supported Hamiltonian isotopy:
We say in such a case that cannot be displaced from . If cannot be displaced from itself we call it nondisplaceable. These properties become especially interesting and purely symplectic when can be displaced from itself or from by a (compactly supported) smooth isotopy.
One of the main themes of the present paper is that “some nondisplaceable sets are more rigid than others.” To explain this, we need the following ramifications of the notion of a nondisplaceable set:
Strong nondisplaceability: A subset is called strongly nondisplaceable if one cannot displace it by any (not necessarily Hamiltonian) symplectomorphism of .
Stable nondisplaceability: Consider with the coordinates and the symplectic form . We say that is stably nondisplaceable if is nondisplaceable in equipped with the split symplectic form . Let us mention that detecting stably nondisplaceable subsets is useful for studying geometry and dynamics of Hamiltonian flows (see for instance [50] for their role in Hofer’s geometry and [51] for their appearance in the context of kick stability in Hamiltonian dynamics).
Formally speaking, the properties of strong and stable nondisplaceability are mutually independent and both are strictly stronger than displaceability.
In the present paper we refine the machinery of partial symplectic quasistates introduced in [23] and get new examples of stably nondisplaceable sets, including certain fibers of moment maps of Hamiltonian torus actions as well as monotone Lagrangian submanifolds discussed by Albers [2] and BiranCornea [15]. Further, we address the following question: given the class of stably nondisplaceable sets, can one distinguish those of them which are also strongly nondisplaceable by means of the Floer theory? Or, other way around, what are the Floerhomological features of stably nondisplaceable but strongly displaceable sets? Toy examples are given by the equator of the symplectic twosphere and by the meridian on a symplectic twotorus. Both are stably nondisplaceable since their Lagrangian Floer homologies are nontrivial. On the other hand, the equator is strongly nondisplaceable, while the meridian is strongly displaceable by a nonHamiltonian shift. Later on we shall explain the difference between these two examples from the viewpoint of Hamiltonian Floer homology and present various generalizations.
The question on Floerhomological characterization of (strongly) nondisplaceable but stably displaceable sets is totally open, see Section 1.7.1 below for an example involving Gromov’s packing theorem and discussion.
Leaving Floertheoretical considerations for the next section, let us outline (in parts, informally) the general scheme of our results: Given a symplectic manifold , we shall define (in the language of the Floer theory) two collections of closed subsets of , heavy subsets and superheavy subsets. Every superheavy subset is heavy, but, in general, not vice versa. Formally speaking, the hierarchy heavysuperheavy depends in a delicate way on the choice of an idempotent in the quantum homology ring of . This and other nuances will be ignored in this outline. The key properties of these collections are as follows (see Theorems 1.2 and 1.5 below):
Invariance: Both collections are invariant under the group of all symplectomorphisms of .
Stable nondisplaceability: Every heavy subset is stably nondisplaceable.
Intersections: Every superheavy subset intersects every heavy subset. In particular, superheavy subsets are strongly nondisplaceable. In contrast to this, heavy subsets can be mutually disjoint and strongly displaceable.
Products: Product of any two (super)heavy subsets is (super)heavy.
What is inside the collections? The collections of heavy and superheavy sets include the following examples:
Stable stems: Let be a finitedimensional Poissoncommutative subspace (i.e. any two functions from commute with respect to the Poisson brackets). Let be the moment map: . A nonempty fiber , , is called a stem of (see [23]) if all nonempty fibers with are displaceable and a stable stem if they are stably displaceable. If a subset of is a (stable) stem of a finitedimensional Poissoncommutative subspace of , it will be called just a (stable) stem. Clearly, any stem is a stable stem. The collection of superheavy subsets includes all stable stems (see Theorem 1.6 below). One readily shows that a direct product of stable stems is a stable stem and that the image of a stable stem under any symplectomorphism is again a stable stem.
The following example of a stable stem is borrowed (with a minor modification) from [23]: Let be a closed subset whose complement is a finite disjoint union of stably displaceable sets. Then is a stable stem. For instance, the codimension1 skeleton of a sufficiently fine triangulation of any closed symplectic manifold is a stable stem. Another example is given by the equator of : it divides the sphere into two displaceable open discs and hence is a stable stem. By taking products, one can get more sophisticated examples of stable stems. Already the product of equators of the twospheres gives rise to a Lagrangian Clifford torus in . To prove its rigidity properties (such as stable nondisplaceability) one has to use nontrivial symplectic tools such as Lagrangian Floer homology, see e.g. [44]. Products of the 1skeletons of fine triangulations of the twospheres can be considered as singular Lagrangian submanifolds, an object which is currently out of reach of the Lagrangian Floer theory.
Another example of stable stems comes from Hamiltonian torus actions. Consider an effective Hamiltonian action with the moment map . Assume that is a normalized Hamiltonian, that is for all . A torus action is called compressible if the image of the homomorphism , induced by the action , is a finite group. One can show that for compressible actions the fiber is a stable stem (see Theorem 1.7 below).
Special fibers of Hamiltonian torus actions: Consider an effective Hamiltonian torus action on a spherically monotone symplectic manifold. Let be the mixed actionMaslov homomorphism introduced in [49]. Since the target space of the moment map is naturally identified with , the pull back of the mixed actionMaslov homomorphism with the reversed sign can be considered as a point of . The preimage is called the special fiber of the action. We shall see below that the special fiber is always nonempty. For monotone symplectic toric manifolds (that is when ) the special fiber is a monotone Lagrangian torus. Note that when the action is compressible we have and therefore the special fiber is a stable stem according to the previous example. It is unknown whether the latter property persists for general noncompressible actions. Thus in what follows we treat stable stems and special fibers as separate examples. The collection of superheavy subsets includes all special fibers (see Theorem 1.9 below).
For instance, consider and the Lagrangian Clifford torus in it (i.e. the torus ). Take the standard Hamiltonian action on preserving the Clifford torus. It has three global fixed points away from the Clifford torus. Make an equivariant symplectic blowup, , of at of these fixed points, , so that the obtained symplectic manifold is spherically monotone. The torus action lifts to a Hamiltonian action on . One can show that its special fiber is the proper transform of the Clifford torus.
Monotone Lagrangian submanifolds: Let be a spherically monotone symplectic manifold, and let be a closed monotone Lagrangian submanifold with the minimal Maslov number . We say that satisfies the Albers condition [2] if the image of the natural morphism contains a nonzero element with
The collection of heavy sets includes all closed monotone Lagrangian submanifolds satisfying the Albers condition (see Theorem 1.15 below).
Specific examples include the meridian on , and all Lagrangian spheres in complex projective hypersurfaces of degree in with . In the case when the fundamental class of divides a nontrivial idempotent in the quantum homology algebra of , is, in fact, superheavy (see Theorem 1.18 below). For instance, this is the case for . Furthermore, a version of superheaviness holds for any Lagrangian sphere in the complex quadric of even (complex) dimension.
However, there exist examples of heavy, but not superheavy, Lagrangian submanifolds: For instance, the meridian of the 2torus is strongly displaceable by a (nonHamiltonian!) shift and hence is not superheavy. Another example of heavy but not superheavy Lagrangian submanifold is the sphere arising as the real part of the Fermat hypersurface
with even and . We refer to Section 1.5 for more details on (super)heavy monotone Lagrangian submanifolds.
Motivation: Our motivation for the selection of examples appearing in the list above is as follows. Stable stems provide a playground for studying symplectic rigidity of singular subsets. In particular, no visible analogue of the conventional Lagrangian Floer homology technique is applicable to them.
Detecting (stable) nondisplaceability of Lagrangian submanifolds via Lagrangian Floer homology is one of the central themes of symplectic topology. In contrast to this, detecting strong nondisplaceabilty has at the moment the status of art rather than science. That’s why we were intrigued by Albers’ observation that monotone Lagrangian submanifolds satisfying his condition are in some situations strongly nondisplaceable. In the present work we tried to digest Albers’ results [2] and look at them from the viewpoint of theory of partial symplectic quasistates developed in [23]. In addition, our result on superheaviness of the Lagrangian antidiagonal in allows us to detect an “exotic” monotone Lagrangian torus in this symplectic manifold: this torus does not intersect the antidiagonal, and hence is not heavy in contrast to the standard Clifford torus, see Example 1.20 below.
In [23] we proved a theorem which roughly speaking states that every (singular) coisotropic foliation has at least one nondisplaceable fiber. However, our proof is nonconstructive and does not tell us which specific fibers are nondisplaceable. The notion of the special fiber arose as an attempt to solve this problem for Hamiltonian circle actions.
Let us mention also that the product property enables us to produce even more examples of (super)heavy subsets by taking products of the subsets appearing in the list.
A few comments on the methods involved into our study of heavy and superheavy subsets are in order. These collections are defined in terms of partial symplectic quasistates which were introduced in [23]. These are certain realvalued functionals on with rich algebraic properties which are constructed by means of the Hamiltonian Floer theory and which conveniently encode a part of the information contained in this theory. In general, the definition of a partial symplectic quasistate involves the choice of an idempotent element in the commutative part of the quantum homology algebra of . Though the default choice is just the unity of the algebra, there exist some other meaningful choices, in particular in the case when is semisimple. This gives rise to another theme discussed in this paper: “visible” topological obstructions to semisimplicity (see Corollary 1.24 and Theorem 1.25 below). For instance, we shall show that if a monotone symplectic manifold contains “too many” disjoint monotone Lagrangian spheres whose minimal Maslov numbers exceed , the quantum homology cannot be semisimple.
Let us pass to the precise setup. For the reader’s convenience, the material presented in this brief outline will be repeated in parts in the next sections in a less compressed form.
1.2 Preliminaries on quantum homology
The Novikov Ring: Let denote a base field which in our case will be either or , and let be a countable subgroup (with respect to the addition). Let be formal variables. Define a field whose elements are generalized Laurent series in of the following form:
Define a ring as the ring of polynomials in with coefficients in . We turn into a graded ring by setting the degree of to be and the degree of to be .
The ring serves as an abstract model of the Novikov ring associated to a symplectic manifold. Let be a closed connected symplectic manifold. Denote by the subgroup of spherical homology classes in the integral homology group . Abusing the notation we will write , for the results of evaluation of the cohomology classes and on . Set
where by definition
Denote by the subgroup of periods of the symplectic form on on spherical homology classes. By definition, the Novikov ring of a symplectic manifold is . In what follows, when is fixed, we abbreviate and write , and instead of , and respectively.
Quantum homology: Set . The quantum homology is defined as follows. First, it is a graded module over given by
with the grading defined by the gradings on and :
Second, and most important, is equipped with a quantum product: if , , their quantum product is a class , defined by
where is defined by the requirement
Here stands for the intersection index and denotes the GromovWitten invariant which, roughly speaking, counts the number of pseudoholomorphic spheres in in the class that meet cycles representing (see [55], [56], [41] for the precise definition).
Extending this definition by linearity to the whole one gets a correctly defined gradedcommutative associative product operation on which is a deformation of the classical product in singular homology [37], [41], [55], [56], [69]. The quantum homology algebra is a ring whose unity is the fundamental class and which is a module of finite rank over . If have graded degrees , then
(1) 
We will be mostly interested in the commutative part of the quantum homology ring (which in the case is, of course, the whole quantum homology ring). For this purpose we introduce the following notation:
We denote by the whole quantum homology if and the evendegree part of if .
In general, given a topological space , we denote by the whole singular homology group if and the evendegree part of if .
Thus, in our notation the ring is always a commutative subring with unity of and a module of finite rank over . We will identify with a subring of by .
1.3 An hierarchy of rigid subsets within Floer theory
Fix a nonzero idempotent (by obvious grading considerations the degree of every idempotent equals ). We shall deal with spectral invariants , where , , is a smooth timedependent and 1periodic in time Hamiltonian function on , or , where is an element of the universal cover of represented by an identitybased path given by the time1 Hamiltonian flow generated by . If is normalized, meaning that for all , then . These invariants, which nowadays are standard objects of the Floer theory, were introduced in [45] (cf. [59] in the aspherical case; also see [42],[43] for an earlier version of the construction and [22] for a summary of definitions and results in the monotone case).
Disclaimer: Throughout the paper we tacitly assume that (as well as , when we speak of stable displaceability) belongs to the class of closed symplectic manifolds for which the spectral invariants are well defined and enjoy the standard list of properties (see e.g. [41, Theorem 12.4.4]). For instance, contains all symplectically aspherical and spherically monotone manifolds. Furthermore, contains all symplectic manifolds for which, on one hand, either or the minimal Chern number (on ) is at least and, on the other hand, is a discrete subgroup of (cf. [64]). The general belief is that the class includes all symplectic manifolds.
Define a functional by
(2) 
It is shown in [23] that the functional has some very special algebraic properties (see Theorem 3.6) which form the axioms of a partial symplectic quasistate introduced in [23]. The next definition is motivated in part by the work of Albers [2].
Definition 1.1.
A closed subset is called heavy (with respect to or with respect to used to define ) if
(3) 
and is called superheavy (with respect to or ) if
(4) 
The default choice of an idempotent is the unity . In this case, as we shall see below, the collections of heavy and superheavy sets satisfy the properties listed in Section 1.1 and include the examples therein. In view of potential applications (including geometric obstructions to semisimplicity of the quantum homology), we shall work, whenever possible, with general idempotents.
The asymmetry between and is related to the fact that the spectral numbers satisfy a triangle inequality , while there may not be a suitable inequality “in the opposite direction”. In the case when such an “opposite” inequality exists (e.g. when is an idempotent and defined by it is a genuine symplectic quasistate – see Section 1.6 below) the symmetry between and gets restored and the classes of heavy and superheavy sets coincide.
Let us emphasize that the notion of (super)heaviness depends on the choice of a coefficient ring for the Floer theory. In this paper the coefficients for the Floer theory will be either or depending on the situation. Unless otherwise stated, our results on (super)heavy subsets are valid for any choice the coefficients.
The group of all symplectomorphisms of acts naturally on and hence on . Clearly, the identity component of acts trivially on and hence for any idempotent the corresponding is invariant. Thus the image of a (super)heavy set under an element of is again a (super)heavy set with respect to the same idempotent . If is invariant under the action of the whole (for instance, if ) the classes of heavy and superheavy sets with respect to are invariant under the action of the whole in agreement with the invariance property presented in Section 1.1 above.
Let us mention also that the collections of (super)heavy sets enjoy a stability property under inclusions: If , , are closed subsets of and is heavy (respectively, superheavy) with respect to an idempotent then is also heavy (respectively, superheavy) with respect to the same .
We are ready now to formulate the main results of the present section.
Theorem 1.2.
Assume and are fixed. Then

Every superheavy set is heavy, but, in general, not vice versa.

Every heavy subset is stably nondisplaceable.

Every superheavy set intersects every heavy set. In particular, a superheavy set cannot be displaced by a symplectic (not necessarily Hamiltonian) isotopy and if the idempotent is invariant under the symplectomorphism group of (e.g. if ), every superheavy set is strongly nondisplaceable.
The following theorem discusses the relation between heaviness/superheaviness properties with respect to different idempotents. In particular, it shows that plays a special role among all the other nonzero idempotents in .
Theorem 1.3.
Assume is a nonzero idempotent in the quantum homology. Then

Every set that is superheavy with respect to is also superheavy with respect to .

Every set that is heavy with respect to is also heavy with respect to .

Assume that the idempotent is a sum of nonzero idempotents and assume that a closed subset is heavy with respect to . Then is heavy with respect to for at least one .
The next proposition shows that, in general, the heaviness of a set does depend on the choice of an idempotent in the quantum homology.
Proposition 1.4.
Consider the torus equipped with the standard symplectic structure . Let be a symplectic blowup of at one point (the blow up is performed in a small ball around the point). Assume that the Lagrangian torus given by does not intersect the ball in , where the blow up was performed.
Then the proper transform of (identified with ) is a Lagrangian submanifold of , which is not heavy with respect to some nonzero idempotent but heavy with respect to . (Here we work with ).
Next, consider direct products of (super)heavy sets. We start with the following convention on tensor products. Let , , be two countable subgroups of . Let be a module over . We put
(5) 
If , are also rings we automatically assume that the middle tensor product is the tensor product of rings. In simple words, we extend both modules to modules and consider the usual tensor product over .
Given two symplectic manifolds, and , note that the subgroups of periods of the symplectic forms satisfy
Furthermore, due to the Künneth formula for quantum homology (see e.g. [41, Exercise 11.1.15] for the statement in the monotone case; the general case in our algebraic setup can be treated similarly) there exists a natural ring monomorphism linear over
We shall fix a pair of idempotents , . The notions of (super)heaviness in and are understood in the sense of idempotents and respectively.
Theorem 1.5.
Assume that is a heavy (resp. superheavy) subset of with respect to some idempotent , . Then the product is a heavy (resp. superheavy) subset of with respect to the idempotent .
An important class of superheavy sets is given by stable stems introduced and illustrated in Section 1.1.
Theorem 1.6.
Every stable stem is a superheavy subset with respect to any nonzero idempotent . In particular, it is strongly and stably nondisplaceable.
In the next section we present an example of stable stems coming from Hamiltonian torus actions.
1.4 Hamiltonian torus actions
Fibers of the moment maps of Hamiltonian torus actions form an interesting playground for testing the various notions of displaceability and heaviness introduced above. Throughout the paper we deal with effective actions only, that is we assume that the map defining the action is a monomorphism. Furthermore, we assume that the moment map of the action is normalized: is a normalized Hamiltonian for all . By the AtiyahGuilleminSternberg theorem [6], [30], the image of is a dimensional convex polytope, called the moment polytope. The subsets , are called fibers of the moment map. A torus action is called compressible if the image of the homomorphism , induced by the action , is a finite group.
Theorem 1.7.
Assume that is equipped with a compressible Hamiltonian action with moment map and moment polytope . Let be any closed convex subset which does not contain . Then the subset is stably displaceable. In particular, the fiber is a stable stem.
Note that for symplectic toric manifolds, that is when , the point is the barycenter of the moment polytope with respect to the Lebesgue measure. This follows from our assumption on the normalization of the moment map.
Theorems 1.6 and 1.7 imply that the fiber of a compressible torus action is stably nondisplaceable, and thus we get the complete description of stably displaceable fibers for such actions.
In the case when the action is not compressible, the question of the complete description of stably nondisplaceable fibers remains open. We make a partial progress in this direction by presenting at least one such fiber, called the special fiber, explicitly in the case when is spherically monotone:
The special fiber can be described via the mixed actionMaslov homomorphism introduced in [49]: Let be a spherically monotone symplectic manifold, and let , be any loop of Hamiltonian diffeomorphisms, with , generated by a 1periodic normalized Hamiltonian function . The orbits of any Hamiltonian loop are contractible due to the standard Floer theory^{1}^{1}1The Floer theory guarantees the existence of at least one contractible periodic orbit – this is not obvious a priori if is not an autonomous flow. Since all the orbits of are homotopic, all of them are contractible.. Pick any point and any disc spanning the orbit . Define the action^{2}^{2}2Note that our action functional and the one in [49] are of opposite signs. of the orbit by
Trivialize the symplectic vector bundle over and denote by the Maslov index of the loop of symplectic matrices corresponding to with respect to the chosen trivialization. One readily checks that, in view of the spherical monotonicity, the quantity
does not depend on the choice of the point and the disc , and is invariant under homotopies of the Hamiltonian loop . In fact, is a well defined homomorphism from to (see [49], [68]).
Assume again that is a Hamiltonian torus action. Write for the induced homomorphism of the fundamental groups. Since the target space of the moment map is naturally identified with , the pull back of the mixed actionMaslov homomorphism with the reversed sign can be considered as a point of . We call it a special point and denote by . The preimage is called the special fiber of the moment map. In the case , when is a realvalued function on , we will call the special value of .
If and is a symplectic toric manifold, then can be defined in purely combinatorial terms involving only the polytope . Namely, pick a vertex of . Since in this case is a Delzant polytope [20], there is a unique (up to a permutation) choice of vectors which

originate at ;

span the rays containing the edges of adjacent to ;

form a basis of over .
Proposition 1.8.
(6) 
Proof.
The vertices of the moment polytope are in onetoone correspondence with the fixed points of the action. Let be the fixed point corresponding to the vertex . Then the vectors , , are simply the weights of the isotropy action on . Since the definition of the mixed actionMaslov invariant of a Hamiltonian circle action does not depend on the choice of a 1periodic orbit and a disc spanning it, let us compute all , , using the constant periodic orbit concentrated at the fixed point and the constant disc spanning it. Clearly,
which readily yields formula (6). ∎
E.Shelukhin pointed out to us that by summing up equations (6) over all the vertices of the moment polytope, one readily gets that .
Theorem 1.9.
Assume is a spherically monotone symplectic manifold equipped with a Hamiltonian action. Then the special fiber of the moment map is superheavy with respect to any (nonzero) idempotent . In particular, it is stably and strongly nondisplaceable.
Let us mention that, in particular, the special fiber is nonempty and so . Moreover is an interior point of – otherwise is isotropic of dimension and hence displaceable (see e.g. [9]).
Remark 1.10.
If (that is we deal with a symplectic toric manifold), the special fiber, say , is a Lagrangian torus. In fact, this torus is monotone: for every we have
where stands for the Maslov class of . This is an immediate consequence of the definitions.
Remark 1.11.
Example 1.12.
Let be the monotone symplectic blow up of at points () which is equivariant with respect to the standard action and which is performed away from the Clifford torus in . Since the blowup is equivariant, comes equipped with a Hamiltonian action extending the action on . The Clifford torus is a fiber of the moment map of the action on . Let be the Lagrangian torus which is the proper transform of the Clifford torus under the blowup – it is a fiber of the moment map of the action on . Using Proposition 1.8 it is easy to see that is the special fiber of . According to Theorem 1.9, it is stably and strongly nondisplaceable. In fact, it is a stem: the displaceability of all the other fibers was checked for in [10], for in [23] and for in [40].
We refer to Section 1.7.2 for further discussion of related problems and very recent advances.
Digression: Calabi vs. actionMaslov. The method used to prove Theorem 1.9 also allows to prove the following result involving the mixed actionMaslov homomorphism. Denote by the symplectic volume of . Consider the function defined by
In the case when is the unity in a field that is a direct summand in the decomposition of the algebra , as an algebra, into a direct sum of subalgebras, is a homogeneous quasimorphism on called Calabi quasimorphism [22],[24],[46]; in the general case it has weaker properties [23]. With this language the functional (on normalized functions) is induced (up to a constant factor) by the pullback of to the Lie algebra of .
Following P.Seidel we described in [22] the restriction of (in fact, for any spherically monotone ) on in terms of the Seidel homomorphism , where denotes the group of invertible elements in the ring . Here we give an alternative description of in terms of the mixed actionMaslov homomorphism which, in turn, also provides certain information about the Seidel homomorphism.
Theorem 1.13.
Assume is spherically monotone and let be defined as above for some nonzero idempotent . Then
Note that, in particular, does not depend on used to define . The theorem also implies that descends to a quasimorphism on if and only if vanishes identically (since descends to a quasimorphism on if and only if – see e.g. [22], Prop. 3.4). The proof of the theorem is given in Section 9.1.
Let us mention also that, interestingly enough, the homomorphism coincides with the restriction to of yet another quasimorphism on constructed by P.Py (see [52, 53]).
Digression: ActionMaslov homomorphism and Futaki invariant. This remark grew from an observation pointed out to us by Chris Woodward – we are grateful to him for that. Assume that our symplectic manifold is complex Kähler (i.e. the symplectic structure on is induced by the Kähler one) and Fano (by this we mean here that ). Assume also that a Hamiltonian action preserves the Kähler metric and the complex structure. For instance, if is a symplectic toric manifold it can be equipped canonically with a complex structure and a Kähler metric invariant under the action on , hence under the action of any subgroup of .
Let be the Hamiltonian vector field generating the Hamiltonian flow . Since preserves the complex structure, one can associate to its Futaki invariant [29]. It has been checked by E.Shelukhin [63] that, up to a universal constant factor, this Futaki invariant is equal to the value of the mixed actionMaslov homomorphism on the loop :
Note that if such an admits a KählerEinstein metric then the Futaki invariant has to vanish [29] – thus if the manifold does not admit a KählerEinstein metric. Moreover, if is toric the opposite is also true: if the Futaki invariant vanishes for any generating a subgroup of the torus acting on then admits a KählerEinstein metric – this follows from a theorem by Wang and Zhu [67], combined with a previous result of Mabuchi [38]. In terms of the moment polytope, the vanishing of the Futaki invariant, and accordingly the existence of a KählerEinstein metric, on a Kähler Fano toric manifold means precisely that the special point of the polytope coincides with the barycenter.
1.5 Super(heavy) monotone Lagrangian submanifolds
Let be a closed spherically monotone symplectic manifold with on , . Let be a closed monotone Lagrangian submanifold with the minimal Maslov number . As usually, we put if . As before, we work with the basic field which is either or . In the case , we assume that is relatively spin, that is is orientable and the 2nd StiefelWhitney class of is the restriction of some integral cohomology class of .
Disclaimer: In the case the results of this section are conditional: We take for granted that Proposition 8.1 below, which was proved by Biran and Cornea [15] for homologies with coefficients, extends to homologies with coefficients. In each of the specific examples below we will explicitly state which we are using and whenever we use we assume that is relatively spin.
Denote by the natural morphism . We say that satisfies the Albers condition [2] if there exists an element so that and
We shall refer to such as to an Albers element of .
Example 1.14.
Assume and is nonzero. This means precisely that is an Albers element of .
A closed monotone Lagrangian submanifold which satisfies this condition (and whose minimal Maslov number is greater than ) will be called homologically nontrivial in .
Theorem 1.15.
Let be a closed monotone Lagrangian submanifold satisfying the Albers condition. Then is heavy with respect to . In particular, any homologically nontrivial Lagrangian submanifold is heavy with respect to .
Example 1.16.
Assume that . Then the homology class of a point is an Albers element of , and hence is heavy. Note that in this case heaviness cannot be improved to superheaviness: the meridian on the twotorus is heavy but not superheavy. Here we took .
Example 1.17 (Lagrangian spheres in Fermat hypersurfaces).
More examples of heavy (but not necessarily superheavy) monotone Lagrangian submanifolds can be constructed as follows^{3}^{3}3We thank P.Biran for his indispensable help with these examples..
Let be a smooth complex hypersurface of degree . The pullback of the standard symplectic structure from turns into a symplectic manifold (of real dimension ). If , then, as it is explained, for instance, in [12], contains a Lagrangian sphere: can be included into a family of algebraic hypersurfaces of with quadratic degenerations at isolated points and the vanishing cycle of such a degeneration can be realized by a Lagrangian sphere following [5], [21], [60], [61], [62].
Let be a projective hypersurface of degree , . The minimal Chern number of equals . Let be a simply connected Lagrangian submanifold (for instance, a Lagrangian sphere).
First, consider the case when is even, is relatively spin and the Euler characteristics of does not vanish (this is the case for a sphere). Then the homology class is nonzero: its selfintersection number in up to the sign equals the Euler characteristic. Thus is an Albers element. (Here we use ). In view of Theorem 1.15, is heavy with respect to .
Second, suppose that is of arbitrary parity but , and no restriction on the Euler characteristics of is assumed anymore. This yields and thus satisfies the Albers condition with the class of a point as an Albers element. Thus is heavy with respect to – here we use .
Finally, fix and an even number such that . Consider a Fermat hypersurface of degree
Its real part lies in the affine chart and is given by the equation
where Since is even, is an dimensional sphere. As it was explained above, is heavy with respect to if either is even (and ) or (and ). However, in either case is not superheavy with respect to . Indeed, let be the group of complex roots of unity. Given a vector , denote by the symplectomorphism of given by
(7) 
If all , then whenever , and thus . Therefore is strongly displaceable and the claim follows from the part (iii) of Theorem 1.2.
The next result gives a userfriendly sufficient condition of superheaviness.
Theorem 1.18.
Assume is homologically nontrivial in and assume is a nonzero idempotent divisible by in , that is . Then is superheavy with respect to .
The homological nontriviality of in the hypothesis of the theorem means just that is an Albers element of (see Example 1.14). In fact, the theorem can be generalized to the cases when has other Albers elements – see Remark 8.3 (ii).
Example 1.19 (Lagrangian spheres in quadrics).
Here we work with . Let be the real part of the Fermat quadric . Assume that is even and is a simply connected Lagrangian submanifold with nonvanishing Euler characteristic (e.g. a Lagrangian sphere). Under this assumption, and , since has nonvanishing selfintersection. Denote by the class of a point. The quantum homology ring of was described by Beauville in [8]. In particular, , where . Thus
are idempotents. One can show that divides and hence is superheavy. Since is invariant under the action of , the manifold is strongly nondisplaceable.
For simplicity, we present the calculation in the case – the general case is absolutely analogous. The 2dimensional quadric is symplectomorphic to . Denote by and the classes of and respectively. Since the symplectic form vanishes on we get that with . It is known that and . Thus , that is divides .
In particular, the Lagrangian antidiagonal
which is diffeomorphic to the sphere, is superheavy with respect to . It is unknown whether is superheavy with respect to . Further information on superheavy Lagrangian submanifolds in the quadrics can be extracted from [15].
Example 1.20 (A nonheavy monotone Lagrangian torus in ).
Consider the quadric from the previous example. We will think of as of the unit sphere in whose symplectic form is the area form divided by . We will work again with . Interestingly enough, such an contains a monotone Lagrangian torus that is not heavy with respect to .
Namely, consider a submanifold given by equations^{4}^{4}4We thank Frol Zapolsky for his help with calculations in this example.
One readily checks that is a monotone Lagrangian torus with which represents a zero element in (both with and ). Thus does not contain any Albers element. Furthermore, is disjoint from the Lagrangian antidiagonal and hence is not heavy with respect to since, as it was shown above, is superheavy with respect to . In particular, is an exotic monotone torus: it is not symplectomorphic to the Clifford torus which is a stem and hence superheavy. A further study of exotic tori in products of spheres is currently being carried out by Y.Chekanov and F.Schlenk.
It is an interesting problem to understand whether is superheavy with respect to , or at least nondisplaceable. Identify with the unit coball bundle of the 2sphere. After such an identification corresponds to the zero section, while corresponds to a monotone Lagrangian torus, say . Interestingly enough, the Lagrangian Floer homology of in (with ) does not vanish as was shown by Albers and Frauenfelder in [3], and thus is not displaceable in . Thus the question on (non)displaceability of is related to understanding of the effect of the compactification of the unit coball bundle to .
The proofs of theorems above are based on spectral estimates due to Albers [2] and BiranCornea [15]. Furthermore, the results above admit various generalizations in the framework of BiranCornea theory of quantum invariants for monotone Lagrangian submanifolds, see [15] and the discussion in Section 8 below.
1.6 An effect of semisimplicity
Recall that a commutative (finitedimensional) algebra over a field is called semisimple if it splits into a direct sum of fields as follows: , where

each is a finitedimensional linear subspace over ;

each is a field with respect to the induced ring structure;

the multiplication in respects the splitting:
A classical theorem of Wedderburn (see e.g. [66], §96) implies that the semisimplicity is equivalent to the absence of nilpotents in the algebra.
Remark 1.21.
Assume that the algebra splits, as an algebra, into a direct sum of two algebras, at least one of which is a field, and let be the unity in that field. In particular, this is the case when is semisimple and is the unity in one of the fields . A slight generalization of the argument in [23, 46] (see [24], the remark on pp. 5657) shows that the partial quasistate associated to is homogeneous (and not just homogeneous as in the general case).
This immediately yields that every set which is heavy with respect to is automatically superheavy with respect to .
In fact, in this situation is a genuine symplectic quasistate in the sense of [23] and, in particular, a topological quasistate in the sense of Aarnes [1] (see [23] for details). In [1] Aarnes proved an analogue of the Riesz representation theorem for topological quasistates which generalizes the correspondence between genuine states (that is positive linear functionals on ) and measures. The object corresponding to a quasistate is called a quasimeasure (or a topological measure). With this language in place, the sets that are (super)heavy with respect to are nothing else but the closed sets of the full quasimeasure . Any two such sets have to intersect for the following basic reason: any quasimeasure is finitely additive on disjoint closed subsets and therefore if two closed subsets of of the full quasimeasure do not intersect, the quasimeasure of their union must be greater than the total quasimeasure of , which is impossible.
Example 1.22.
In this example we again assume that . Let be equipped with the FubiniStudy symplectic structure , normalized so that , and let be the homology class of the hyperplane. One readily verifies the following algebra isomorphism
where
is the field of Laurenttype series in with coefficients in and . Since no root of degree or more of is contained in , the polynomial is irreducible over for any (see e.g. [34], Theorem 9.1) and therefore is a field. Hence the collections of heavy and superheavy sets with respect to the fundamental class coincide.
We claim that is superheavy. The case corresponds to the equator of the sphere, which is known to be a stable stem. For , note that and is an Albers element of . Therefore, is heavy by Theorem 1.15, and hence superheavy.
Theorem 1.23.
Assume that is semisimple and splits into a direct sum of fields whose unities will be denoted by . Assume that a closed subset is heavy with respect to a nonzero idempotent – as one can easily see, such an idempotent has to be of the form for some . Then is superheavy with respect to some , .
The theorem yields the following geometric characterization of nonsemisimplicity of . Namely, define the symplectic Torelli group as the group of all symplectomorphisms of which induce the identity map on . For instance, this group contains . Note that any element of the symplectic Torelli group acts trivially on the quantum homology of and hence maps sets (super)heavy with respect to an idempotent to sets (super)heavy with respect to .
Now Theorem 1.23 readily implies the following
Corollary 1.24.
Assume that contains a closed subset which is heavy with respect to a nonzero idempotent and displaceable by a symplectomorphism from the symplectic Torelli group. Then