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An extension of the Siegel space of complex abelian varieties and conjectures on stability structures

We study semi--algebraic domains associated with symplectic tori and conjecturally identified with spaces of stability conditions on the Fukaya categories of these tori. Our motivation is to test which results from the theory of flat surfaces could hold for more general spaces of stability conditions. The main results concern systolic bounds and volume of the moduli space.


Introduction
The theory of quadratic differentials on Riemann surfaces has seen surprising connections with a number of different areas of mathematics. From a geometric point of view they give flat surfaces with conical singular points whose cone angles are integer multiples of π. We refer to [29] for a broad introduction. A particularly fundamental result, due to Masur [21] and Veech [28], is the finiteness of the volume of the moduli space of flat surfaces of given type, which allows methods from ergodic theory to come into play.
More recently, motivated partly by considerations in string theory, moduli spaces of flat surfaces were shown to be special instances of spaces of stability conditions, as defined by Bridgeland, in works by Bridgeland-Smith [8], and Kontsevich, Katzarkov, and the author [12]. In the latter work [12], a triangulated category, the partially wrapped Fukaya category F (S), is defined for surfaces S with markings. It is then shown that a flat metric on S induces a stability condition on F (S) whose stable objects correspond to (finite length) geodesics on S. In this way one obtains an embedding of the space of flat metrics on S, M(S), into the space of stability conditions, Stab(F (S)), of F (S) as a union of connected components of the latter. Moreover, all of the important features of M(S), such as the period map and the wall-and-chamber structure, can be defined intrinsically on Stab(F (S)) starting from the axiomatics of a stability condition.
Given these results it is natural to wonder how much of the theory of flat surfaces survives for spaces of stability conditions on more general categories, e.g.
Fukaya categories, F (M), of higher-dimensional symplectic manifolds, M. See also the discussion by Smith [26]. The case when dim R M = 6 and M is compact is of particular interest in view of the theory of categorical Donaldson-Thomas invariants [17]. An immediate difficulty is that constructing stability conditions and determining the geometry of Stab(C) is a hard problem in higher dimensions, though progress has been made [3,20,2].
In the case when M is a torus with rational constant symplectic form, there is an explicit conjectural description of Stab(F (M)) due to Kontsevich [16]. It involves the following generalization of the notion of a compatible complex structure on a symplectic vector space, (V, ω). Suppose dim R V = 2n and let U(V ) be the space of C-valued alternating n-forms on V which are primitive (i.e. Ω ∧ ω = 0) and whose restriction to any Lagrangian subspace L ⊂ V is non-vanishing (as a top-degree form on L). It turns out that U(V ) has two connected components U ± (V ) of which U + (V ) contains forms which are of type (n, 0) with respect to some compatible complex structure on V . Some first results about U(V ) are collected in the following theorem. See Propositions 2.3, 2.5, and 2.12 in the main text. We also determine the structure of U + (R 2 ) and U + (R 4 ) completely, see Subsection 2.4, and classify el-ements of U + (R 6 ) in terms of pairs of compatible complex structures. This is based on prior work of Hitchin on three-forms [13]. Conjecturally, the universal cover of U + (V ) is (perhaps a component of) the space of stability conditions on the Fukaya category of a torus V /Λ with constant rational symplectic form.
More generally, one may consider a sort of "soft" version of Calabi-Yau geometry given by a symplectic manifold (M, ω) with a closed complex-valued middle degree form Ω such that Ω p ∈ U + (T p M) for all p ∈ M, i.e. which does not vanish on any Lagrangian subspace of any tangent space. Some basic definitions in this direction are collected in Section 3. This type of structure was suggested by Kontsevich as a geometric approach to spaces of stability conditions, in particular as a way of understanding the origin of non-geometric stability conditions. By contrast, a geometric stability condition is one coming from Calabi-Yau geometry by the conjectural construction as described by Joyce [15].
In the final section we study U + (R 2n )/Sp(2n, Z), which generalizes the moduli space of flat tori U + (R 2 )/SL(2, Z) and contains the moduli space of principally polarized abelian varieties. Our first result in this direction concerns the existence of a systolic bound, i.e. an inequality of the form where Sys Ω (T 2n ) is the minimum of the volumes of compact special Lagrangians in the torus and Vol Ω (T 2n ) is the volume computed by integrating the form Ω ∧ Ω. These quantities were considered in the context of Calabi-Yau manifolds and stability conditions by Fan-Kanazawa-Yau [10], and Fan [9] is the subset of forms which are suitably normalized to unit volume.
We believe there there is still much to be said about the spaces U(V ) and open questions are indicated throughout.

Acknowledgements
The author would like to thank Yu-Wei Fan, Maxim Kontsevich, Pranav Pandit, Hiro Lee Tanaka, and Alex Wright for valuable discussions.

Linear theory
In this section we define and study the spaces U(V ) assigned to a symplectic vector

Primitive forms
We recall some standard symplectic linear algebra. Let (V, ω) be a symplectic vector space of dimension 2n. The nilpotent Lefschetz operator L(α) := ω ∧ α extends to a linear representation of the Lie algebra sl(2) on exterior forms, Λ • V ∨ .
Concretely, this implies that any k-form, α, on V has a unique decomposition is non-zero and primitive. Thus, the classical Plücker embedding restricts to an embedding is the set of decomposable k-vectors which is cut out by the quadratic Plücker relations, giving Gr iso (k, V ) the structure of a real algebraic variety.

A non-vanishing condition
Any R-valued n-form, Ω, on a 2n-dimensional symplectic vector space, V , (over R) must vanish on some Lagrangian subspace for topological reasons. On the other hand, if we choose a compatible complex structure on V and let Ω be a non-zero complex valued form of type (n, 0) or (0, n) then for any Lagrangian subspace L ⊂ V . This is easy to see using, for example, the fact that the unitary group, U(V ), acts transitively on the set of Lagrangian subspaces and Ω transforms as a character of U(V ). More generally we consider the following types of forms.
Definition 2.1. A form Ω ∈ Λ n pr V ∨ ⊗ C is non-vanishing on Lagrangian subspaces if its pullback to any Lagrangian subspace L ⊂ V is non-zero. Let which is closed and algebraic in Λ n pr V ∨ ⊗ C × LGr(V ). Its projection to the first factor, pr 1 (Z), is closed by compactness of LGr(V ) and semi-algebraic by the Tarski-Seidenberg theorem (quantifier elimination for semialgebraic sets). The statement follows since As a consequence, U(V ) has the structure of a complex manifold and inherits the indefinite Hermitian metric Also, U(V ) has a natural right action of Sp(V ) and a left action of GL(2, R) coming from its action on C = R 2 , and these actions commute.
However, U(V ) has much larger dimension than Sp(V ) in general, so cannot be a homogeneous space for that group.

Topology of U (n)
The topology of U(n) turns out to be quite simple. We will show in this subsection that for n ≥ 1 there are two connected components, U ± (n), both homotopic to S 1 .
For V = C n the canonical generator is represented by the loop where c 1 , c 2 ∈ C are coefficients of Ω: Thus, depending on the sign of Re(c 1 c 2 ), φ Ω sends the canonical generator either to the positive or negative generator of π 1 (R/πZ).
The map defines a loop of Maslov index 2n in the Lagrangian Grassmannian, so since Ω is positively oriented the map has degree n. Looking at (2.16) this implies that p has degree n with all roots in the open unit disk. In particular, Ω n,0 = 0.
For Ω ∈ U + (V ) consider and we claim that Ω t ∈ U + (V ) for all t ∈ [0, 1]. This follows since if w, p are as above then as p has no zeros outside the unit disk.
We can also say more about the maps φ Ω .

Corollary 2.6.
If Ω ∈ U(V ), then the map φ Ω : LGr is a fiber bundle with fibers diffeomorphic to the special Lagrangian Grassmannian Proof. The explicit formula (2.13) shows that φ Ω has no critical points, thus gives a fiber bundle. For Ω ∈ U(V ) of type (n, 0) or (0, n) it is clear that the fibers are diffeomorphic to SU(n)/SO(n). Since U ± (V ) are path connected, the claim follows.

U(2)
We suppose dim R V = 4, so dim C U(V ) = 5. The 5-dimensional space Λ 2 pr V of primitive bivectors has a non-degenerate symmetric bilinear form of signature We will use it to identify Λ 2 pr V with its dual, Λ 2 pr V ∨ . For V = C 2 an orthonormal basis of this space is given by .

(Plücker relation). Thus, the affine cone over the Lagrangian Grassmannian
LGr(V ) ⊂ P(Λ 2 pr V ) is just the light-cone in Λ 2 pr V . This allows us to give explicit inequalities for U(2).
pr V ∨ ⊗ C is non-vanishing on Lagrangian subspaces if and only if α and β span a spacelike plane in Λ 2 pr V ∨ , i.e. the symmetric matrix Proof. Identify Λ 2 pr V ∼ = Λ 2 pr V ∨ as before and consider α, β as vectors in the former. A Lagrangian subspace, L, is represented by a lightlike bivector v = 0 and Ω vanishes on L iff v is orthogonal to both α and β. Thus, Ω is does not vanish on any Lagrangian subspace iff the orthogonal complement P ⊥ to the subspace P ⊂ Λ 2 pr V spanned by α and β intersects the lightcone in the origin only. This happens iff the restriction of the symmetric form to P ⊥ is definite, which implies dim R P ⊥ = 3 and the symmetric form is negative definite on P ⊥ . Thus dim R P = 2 and the symmetric form is positive definite on P .
It turns out the the matrix S Ω which appeared in the previous proposition is a complete invariant of Ω ∈ U + (V ) under the Sp(V )-action.
Proof. The symplectic group Sp(V ) fixes ω by definition and thus preserves the consequence of the coincidence of root systems B 2 = C 2 . Since SO + (2, 3) acts transitively on spacelike planes in Λ 2 pr V ∨ , the only invariants of Ω are the lengths of α and β, and the angle between them. This is precisely the data recorded by It is an easy consequence of the previous proposition that every Ω ∈ U + (2) can be written as with respect to a suitable symplectic basis. Note that this form is SU(2)-invariant.
Moreover the double coset space reduces to a point. This fails for dim R V > 4 by a dimension argument.

Functoriality
We again start the subsection with a review of a bit of symplectic linear algebra.
The graph of a linear symplectomorphism V 1 → V 2 is a Lagrangian subspace in More generally one considers arbitrary Lagrangian subspaces of V 1 × V 2 with this symplectic form, the linear Lagrangian correspondences, as morphisms in a linear symplectic category. Composition of such morphisms is not continuous on the Lagrangian Grassmannian, and from the modern point of view one should instead use derived intersection, though this will not be necessary for the present discussion. Any linear Lagrangian correspondence factors into a symplectic quotient, a symplectomorphism, and the inverse of a symplectic quotient [4]. A linear symplectic quotient (aka reduction) is a Lagrangian correspondence of the form is the symplectic complement and W is coisotropic iff W ⊥ ⊂ W .
Fix a coisotropic subspace W ⊂ V and a non-zero volume element ν ∈ Λ top W ⊥ , then we get a map Let (V 1 , ω 1 ) and (V 2 , ω 2 ) be symplectic vector spaces of dimensions 2n 1 , 2n 2 respectively, and are canonical projections, and we can consider A natural question at this point is: If Ω k ∈ U(V k ) for k = 1, 2 does it follow that Ω ∈ U(V )? This seems to be a somewhat delicate point and we prove only a partial result. To state it we make the following definition.
Equivalently, there exists a compatible complex structure on V such that Ω is a sum of forms of types (n, 0) and (0, n), 2n = dim R V . Denote by U ag ⊂ U + (V ) the subset of almost geometric forms.
Note that we have inclusions which are all strict for sufficiently large dimension of V . In some respects, the almost geometric forms are a more natural class than the geometric ones, c.f. the remark after Definition 2.1. The following theorem also supports this.
The analogous statement fails if Ω 1 ∈ U + (V ) but Ω 2 ∈ U − (V ), or if Ω 1 and Ω 2 are not required to be primitive forms.
Proof. Let L ⊂ V 1 ⊕V 2 be a Lagrangian subspace, i.e. a Lagrangian correspondence from V 1 to V 2 with the negative symplectic form. By the classification of linear Lagrangian correspondences, the projections C k := p k (L) ⊂ V k are coisotropic and there is a diagram (2.33) where φ is a linear anti-symplectomorphism so that Choose non-zero elements ν k ∈ Det(C ⊥ k ), where Det denotes the top exterior power. In view of the exact sequence the ν k give an identification We claim that if Ω 1 is almost geometric, then so is any reduction of it, in particular ν 1 Ω 1 | C 1 . By induction it suffices to show this if C 1 is a hyperplane and ν 1 a vector. Choosing suitable coordinates we can assume which is almost geometric on C n−1 .
By the preceding arguments we have reduced the problem to showing the following: Let V be a symplectic vector space Ω 1 ∈ U ag (V ), Ω 2 ∈ U + (V ), and φ : V → V an anti-symplectomorphism, then Ω 1 ∧ φ * Ω 2 = 0. Also is suffices to show this for just one particular φ, which we chose to be complex conjugation v →v on V = C n . In suitable coordinates we have with |c 1 | > |c 2 |, |c 3 | > |c 4 |, and writing dZ := dz 1 ∧ . . . ∧ dz n , so which completes the proof.

The action of Sp(V ) on U (V )
The action of Sp(V ) on U(V ) has much better properties than the action on the entire vector space Λ n pr V ∨ ⊗ C, for example it is proper with stable orbits in the sense of geometric invariant theory. We also show in this subsection that U + (V ) fibers Sp(V )-equivariantly over the geometric part U geom (V ). Let X ∈ p, then we may assume that X is diagonal of the form  Next, we study the action of Sp(V ) from the point of view of (real) geometric invariant theory as developped in e.g. [24], [5]. To see this let X ∈ p, then we may assume that X is diagonal of the form
The classification of alternating trilinear forms on a six-dimensional vector space goes back to Reichel [23]. Restricting to primitive forms and the action of Sp(6) ⊂ GL(6), the problem was solved for algebraically closed fields of characteristic zero by Igusa [14]. Over R, the classification is contained in work of Lychagin-Rubtsov-Chekalov [19]. For given α ∈ Λ 3 pr V ∨ the key is to consider the symmetric bilinear form, q α , on V given by In fact the map has an interpretation as a moment map for the Sp(V ) action on Λ 3 pr V ∨ . Since V has a natural volume element, ω 3 /3!, the determinant of q α is a well defined real number, in fact the only continuous invariant of α. Furthermore, the Plücker relations for LGr(V ) are just q α = 0.
We are interested in the case when α is the real or imaginary part of Ω ∈ U(V ), in which case q α turns out to be positive definite. form v α. Any alternating 2-form has even rank, so N must be even-dimensional.
Furthermore v ∈ N, and we claim that N ⊂ V is a symplectic subspace with The classification result of [19] implies that if α, β ∈ Λ 3 pr V ∨ with q α , q β positive definite, then α and β lie in the same Sp(V ) orbit if and only if det(q α ) = det(q β ).
In particular, if q α is positive definite then there is a compatible almost complex structure on V and Ω ∈ Λ 3,0 V ∨ such that α = Re(Ω). We have proven the following. Proposition 2.17. Let dim R V = 6 and Ω ∈ U(V ), then there is a pair J 1 , J 2 of compatible complex structures on V and Ω 1 , If J is any complex structure on C n which is compatible with its standard symplectic structure, then there is a unitary change of coordinates (preserving the standard hermitian structure) such that In these coordinates the 1-forms λ k dx k + idy k are of type (1, 0) with respect to J.
Proof. According to Proposition 2.17 we can write Ω = α + β with α = c 1 dZ +c 1 dZ which is equivalent to Re(c 1c2 ) > 0. On the other hand, This completes the proof since complex conjugation interchanges U ± (3) and the sign of Ω ∧Ω.

Global theory
In this section we define a global version of the structure considered in the previous section. More precisely, we consider symplectic manifolds equipped with a closed and primitive complex-valued middle-degree form, Ω, which satisfies the non-vanishing condition studied in the previous section pointwise. When M is compact we associate with such an Ω a volume and systole (Subsection 3.2). These definitions will be applied to the special case when M is a rational symplectic torus in the next section. Finally, we discuss a relaxed form of the closedness condition on Ω. Although this definition considerably relaxes the Calabi-Yau condition, it is perhaps still too strict and we will discuss potential generalizations later. Here are some examples of C-polarizations.

C-polarizations
1. Let V be a symplectic vector space with lattice Λ ∈ V and Ω ∈ U + (V ), then we can consider Ω as a constant differential form on the symplectic torus V /Λ.

If (M, ω, J) is a Kähler manifold with non-vanishing holomorphic volume
form Ω (not necessarily satisfying the Calabi-Yau condition), then Ω defines a C-polarization. One could attempt to generalize this example by requiring J to be only a compatible almost complex structure, but in fact the condition dΩ = 0 for a non-vanishing (n, 0) form implies integrability of J by the Newlander-Nirenberg theorem.
Associated with a C-polarization Ω on (M, ω) is a central charge Thus, one has all the data required for a stability condition on the Fukaya category, F (M), of M if we define semistable objects to be those which can be represented by an immersed special Lagrangian submanifold. This is not yet a precise definition, since one probably needs to allow singular Lagrangian submanifolds, and it is unknown which singularities to allow and how to include such objects in F (M).
Checking that this data satisfies the axioms of a stability condition is another matter. At least one of the axioms, the support property, is easy. Not coincidentally, the argument below served as motivation for Kontsevich-Soibelman to introduce the support property in [17], see also [18]. For the special case of surfaces it can be found in [12].

Proposition 3.2 (Support property).
Let (X, ω) be a compact symplectic manifold, dim R X = 2n, with C-polarization Ω. Then there is a norm on H n (X; R) and a constant C > 0 such that Z) is the class of a compact special Lagrangian submanifold.
Proof. Let α 1 , . . . , α m be n-forms on X such that [α 1 ], . . . , [α m ] is a basis of H n dR (X; R). Consider the norm on H n (X; R). Since X is compact and Ω is non-vanishing on Lagrangian subspaces, where the last equality follows since L is special.

Volume and systole
Let (M, ω) be a compact symplectic manifold of dimension 2n with C-polarization Ω. We have two natural top-degree forms on M, namely  A, B)) for the Fukaya category of M. This fact was used by Fan-Kanazawa-Yau [10], see also [9], to assign a "volume" to a stability condition on a triangulated category.
Define the systole to be volume of the smallest special Lagrangian submanifold in M, i.e. Calabi-Yau, this type of systole was considered by Fan [9]. There is a reasonable variant of the definition where one considers all compact Lagrangian submanifolds, not just the special ones. We will soon restrict to rational symplectic tori, and since these have a good supply of special Lagrangian subtori, these distinctions will not be important.

Second order condition
We will discuss in this subsection a more general version of the notion of a Cpolarization (Definition 3.1) suggested by a variant of Hodge theory for symplectic manifolds developed by Tseng-Yau [27]. To motivate this generalization we consider the famous Kodaira-Thurston nilmanifold example. Topologically, this is an S 1 bundle over T 3 which can be constructed as a quotient M = R 4 /Z 4 where Z 4 acts by A frame of the cotangent bundle is given by is of type (2, 0) with respect to a compatible almost complex structure. However so Ω is not closed.
To continue we note that Ω is "closed" in a weaker sense: where Λ is the dual Lefschetz operator defined by contraction with ω. This condition ensures that which is the natural home for currents associated with closed Lagrangian submanifolds.
The upshot is that the condition dΩ = 0 can be replaced, for present purposes, by the weaker condition (3.16). Imposing the pointwise condition Ω p | L = 0 for any Lagrangian subspace L ⊂ T p M is justified at least in the case of rational symplectic tori, as will be shown in Proposition 4.1.

Rational symplectic tori
In this section we give an equivalent definition of U(V ) as a space of linear functional Λ n pr V → C satisfying the support property with respect to homology classes of linear Lagrangian tori in a rational symplectic torus. In Subsection 4.2 we will establish a systolic bound on U ag (V ), and in Subsection 4. To be more concrete, we may assume without loss of generality that V = R 2n , ω a rational multiple of the standard symplectic form, and

Support property
Let Γ be a free abelian group of finite rank and S ⊂ Γ a subset, then an additive map Z : Γ → C is said to satisfy the support property (with respect to S) if there is a norm . on Γ ⊗ R and a constant C > 0 such that Note that if Z satisfies this property then Z(S) is a discrete subset of C. The support property was introduced by Kontsevich-Soibelman in the definition of stability data and stability structure in [17], and is usually added to Bridgeland's axioms of a stability condition [6], where S is the set of classes of semistable objects. Proof. One implication is a special case of Proposition 3.2. Suppose instead that the support property holds for some Ω ∈ Λ n pr V ∨ ⊗ C. Thus there is a C > 0 such that for γ ∈ Λ n pr Λ =: Γ the class of a linear Lagrangian subtorus. The right hand side of (4.3) is a well defined continuous function on LGr(V ), and since it is bounded below on a dense subset, it must be non-vanishing everywhere. This proves that Ω does not vanish on any Lagrangian subspace in V .
The previous proposition provides good evidence that where the left hand side is the space of stability conditions on the Fukaya category (over the Novikov field) of a rational symplectic torus, and the right hand side is the universal cover of the K(Z, 1)-space U + (V ). A similar conjecture was suggested by Kontsevich [16].

Systolic bound
Let (M, ω) be a compact symplectic manifold. A systolic bound is an inequality of the form where the constant C > 0 depends only on (M, ω) and not on Ω, which is allowed to vary in some set of C-polarizations (to be specified x ≤ C n n Vol(R n /Λ) for some constant C n > 0 depending only on n, not Λ. In fact, the n = 1 case of Theorem 4.2 is just the n = 2 case of the lattice point theorem.
Proof. It suffices to consider the standard symplectic torus R 2n /Z 2n , since passing to a finite cover can only scale systole and volume by a fixed finite amount.
We will make use of Siegel's description of the space of compatible complex structures on V , see [25]. Recall that the Siegel space is the set S n of matrices Z = X + iY ∈ Mat(n × n, C) with X, Y symmetric and Y positive definite. The group Sp(2n, R) acts transitively on S n via (4.7) The point X + iY ∈ S n corresponds to the Kähler torus R 2n /Λ with which is isomorphic to the standard torus R 2n /Z 2n with complex structure (4.9) Fix a compatible complex structure on V , i.e. a point Z = X + iY ∈ S n , and let  Applying Minkowski's theorem to the lattice Z n with the positive definite quadratic form Y , we can assume, after performing some change of coordinates given by a block-diagonal element of Sp(2n, Z), that (4.11) where Y = (y ij ) 1≤i,j≤n .
Given p, q ∈ Z, not both zero, let to C with the standard metric and the lattice Z ⊕ z 11 Z we find (4.13) Combining the above we get (4.14) Sys The previous theorem can be strengthened to allow also those Ω ∈ U + (V ) which are in the GL + (2, R)-orbit of an (n, 0) form. Proof. We will deal with the cases of odd and even n separately, where 2n = dim V .
Assume now that n is odd, so that Vol Ω (T 2n ) is invariant under the SL(2, R)action. The proof of Theorem 4.2 showed that there is a lattice Z ⊕ τ Z ⊂ C (where τ := z 11 ) all of whose non-zero elements are central charges of (possibly immersed) linear Lagrangian tori in T 2n , and furthermore the area of this lattice, Im(τ ), has an absolute upper bound for fixed Vol Ω (T 2n ). This property persists when applying some element of SL(2, R) to Ω, so again by Minkowski's theorem in the plane we get an upper bound on the smallest volume of a linear Lagrangian subtorus.
As a corollary we find that a systolic bound holds on all of U + (2).

Volume of moduli space
Suppose V is a symplectic vector space of dimension 2n with n ≥ 1 odd, then Λ n pr V ∨ ⊗ C has a natural symplectic structure (4.19) (Ω 1 , Ω 2 ) → Im (Ω 1 ∧ Ω 2 ) ω n n!  which is a very special case of a theorem due to Masur [21] and Veech [28] on the finiteness of the volumes of moduli spaces of flat surfaces.
In contrast, the volume turns out to be infinite for n = 3. Proof. Let V be a 6-dimensional symplectic vector space. Given α ∈ Λ 3 V ∨ we defined in (2.56) a quadratic form q α on V which depends quadratically on α, i.e. q λα = λ 2 q α . Furthermore, if Ω ∈ U(V ) then q ReΩ is positive definite by Lemma 2.16. To get a single number from q α we first define K α : V → V by where C := 12 log 2. This implies that each subset (4.30) f −1 ([kC, (k + 1)C)) ⊂ M 1 (3), k ∈ Z has the same positive volume. Since the whole space is the disjoint union of these subsets, the claim follows.