Systolic inequalities for K3 surfaces via stability conditions

We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface X, there exists a constant C depending only on the rank and discriminant of NS(X), such that sys(σ)2≤C·vol(σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathrm {sys}(\sigma )^2\le C\cdot \mathrm {vol}(\sigma ) \end{aligned}$$\end{document}holds for any stability condition on DbCoh(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}^b\mathrm {Coh}(X)$$\end{document}. This is an algebro-geometric generalization of a classical systolic inequality on two-tori. We also discuss applications of this inequality in symplectic geometry.


Introduction
Let (M, g) be a Riemannian manifold. Its systole sys(M, g) is defined to be the least length of a non-contractible loop in M. In 1949, Charles Loewner proved that sys T 2 , g 2 ≤ 2 √ 3 vol T 2 , g holds for any Riemannian metric g on a two-torus T 2 . There are various generalizations of Loewner's tours systolic inequality. We refer to [16] for a survey on the rich subject of systolic geometry. The first goal of the present article is to propose a new generalization of Loewner's torus systolic inequality from the perspective of Calabi-Yau geometry. We start with an observation in the case of a two-torus. Suppose that the torus is flat T 2 τ ∼ = C/Z + τ Z, and is equipped with the standard complex structure = dz and symplectic structure ω = dx ∧ dy. Then the shortest non-contractible loops must be straight lines, therefore are special Lagrangian submanifolds with respect to the complex and symplectic structures. Under these assumptions, Loewner's torus systolic inequality can be interpreted as: The key observation is that the quantities in both sides of this inequality can be generalized to any Calabi-Yau manifold. We propose the following definition of systole of a Calabi-Yau manifold, with respect to its complex and symplectic structures. With this definition, we propose the following question that naturally generalizes inequality (1) to any Calabi-Yau manifold. Here we treat the Calabi-Yau manifolds topologically so that the complex structures can vary. Note that the choice of the symplectic structure is not important in the case of two-tori, since any one-dimensional submanifold is Lagrangian in T 2 . However, in higher dimensions, the notion of Lagrangian submanifolds certainly depends on the choice of the symplectic structure. Therefore the systolic constant C should depend on the symplectic structure in general, unlike the case of T 2 . Also, note that the ratio | L | 2 / Y ∧ has been considered in the context of attractor mechanism in physics [9,20,23], which is of independent interest.
The second goal of the present article is to introduce the definitions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. Note that in the definition of systole of a Bridgeland stability condition, the minimum can always be attained by some σ -stable object E, therefore we can write "min" instead of "inf" (see Remark 2.5).
The motivations of these definitions stem from the correspondence between flat surfaces and stability conditions, and the conjectural description of stability conditions on the Fukaya categories of Calabi-Yau manifolds. We refer to Sect. 2 for more details. Note that under these correspondences, sys(σ ) is the categorical generalization of sys(Y , ω, ) in Definition 1.1, and vol(σ ) is the categorical generalization of the holomorphic volume Y ∧ .
As a sanity check of these definitions, we prove the following categorical analogue of Loewner's torus systolic inequality.
We then propose the following algebro-geometric analogue of Question 1.2, which is the higher-dimensional generalization of Theorem 1.5.

Question 1.6 Let X be a Calabi-Yau manifold and
be a complex structure on X . Let D = D b Coh(X , ) be its derived category of coherent sheaves. Does there exist a constant C = C(X , ) > 0 such that holds for any σ ∈ Stab † (D)? Here Stab † (D) denotes the distinguished connected component of Stab(D) that contains geometric stability conditions.
Note that Questions 1.2 and 1.6 are related via the mirror symmetry conjecture, which is a conjectural duality between algebraic geometry and symplectic geometry. The homological mirror symmetry conjecture proposed by Kontsevich [18] states that for any Calabi-Yau manifold with a symplectic structure (Y , ω), there exists a Calabi-Yau manifold with a complex structure (X , ) such that there is an equivalence between the derived Fukaya category of Y and the bounded derived category of coherent sheaves on X : It is conjectured by Bridgeland [6] and Joyce [15] that a holomorphic top form on Y should give rise to a Bridgeland stability condition σ on D b Fuk(Y , ω) (see Conjecture 2.3). The conjectural stability condition satisfies sys(σ ) = sys(Y , ω, ) and vol(σ ) = Y ∧ . Therefore we can consider Question 1.6 as the mirror counterpart of Question 1.2. We refer to Sect. 5 for more discussions on this.
The third goal, which is the main result of the present article, is to give an affirmative answer to Question 1.6 for any complex projective K3 surface. A priori there is no reason to believe that Question 1.2 and Question 1.6 have affirmative answers in general. The following theorem is the first evidence that the natural categorical generalization of systolic inequality is possible for higher-dimensional Calabi-Yau manifolds.
Here ρ and disc denote the rank and the discriminant of the Néron-Severi group NS(X ), respectively.
Moreover, when the K3 surface is of Picard rank one, we can use a different method to get a better systolic bound.
Finally, we remark that one can also define a categorical generalization of systole using only the spherical objects (Remark 4.8), which we denoted by sys sph (σ ). However, as we prove in Proposition 4.9, the ratio sys sph (σ ) 2 /vol(σ ) is unbounded in general.

Related work
After the first version of the present article was posted online, Haiden [13] proves a systolic inequality for certain higher-dimensional symplectic torus, and Pacini [25] proposes a higherdimensional generalization of extremal lengths and complex systolic inequalities.

Organization
In Sect. 2, we recall the definition of Bridgeland stability conditions and introduce the notions of categorical systole and categorical volume. In Sect. 3, we give an affirmative answer to Question 1.6 for elliptic curves by proving Theorem 1.5. In Sect. 4, we give an affirmative answer to Question 1.6 for any K3 surface by proving Theorem 1.7. In Sect. 5, we discuss some directions for future studies.

Bridgeland stability conditions
In the seminal work [4], Bridgeland introduced the notion of stability conditions on triangulated categories. We recall the definition and some basic properties of Bridgeland stability conditions.
Throughout the article, a triangulated category D is essentially small, linear over C, and is of finite type. The last condition means that for any pair of objects E, F ∈ D, the C-vector The Euler form χ on the Grothendieck group K (D) is given by the alternating sum  such that: (e) (Support property [19]) There exists a constant C > 0 and a norm || · || on N (D) ⊗ Z R such that
The group homomorphism Z is called the central charge, and the nonzero objects in P(φ) are called the semistable objects of phase φ. The additive subcategories P(φ) actually are abelian, and the simple objects of P(φ) are said to be stable.
The space of (full numerical) Bridgeland stability conditions on D is denoted by Stab(D). There is a nice topology on Stab(D) introduced by Bridgeland, which is induced by the generalized distance: The forgetful map is a local homeomorphism [4,19]. Hence Stab(D) is a complex manifold. There are two natural group actions on the space of Bridgeland stability conditions Stab(D) which commute with each other [4,Lemma 8.2]. Firstly, the group of autoequivalences Aut(D) acts on Stab(D) as isometries with respect to the generalized metric: Let ∈ Aut(D) be an autoequivalence, define where [ ] is the induced automorphism on N (D), and P (φ) := (P(φ)).
Secondly, the universal cover GL + (2, R) also admits a natural group action on Stab(D).

Categorical systoles
We first recall some results and conjectures that motivate our definition of categorical systoles.
In a striking series of work by Gaiotto-Moore-Neitzke [12], Bridgeland-Smith [7] and Haiden-Katzarkov-Kontsevich [14], the connections between stability conditions and Teichmüller theory have been established. One of the main results in this direction is the following theorem. Recall that the systole of a flat surface is defined to be the length of its shortest saddle connection. Based on the correspondence established in Theorem 2.2, it is natural to define the systole of a Bridgeland stability condition to be the smallest absolute value of central charge of semistable objects.
Another important source of motivation for defining categorical systole is a conjectural description of stability conditions on the Fukaya categories of Calabi-Yau manifolds, proposed by Bridgeland and Joyce. If we assume this conjecture to be true, then the definition of systole of (Y , ω, ) in Definition 1.1 can be written as Motivating from the above discussions, we propose the following definition of systole of a Bridgeland stability condition.

Definition 2.4
Let σ be a Bridgeland stability condition on a triangulated category D. Its systole is defined to be sys(σ ) := min{|Z σ (E)|: E is a σ -semistable object in D}.

Remark 2.5
Note that we can write "min" instead of "inf" in the definition of categorical systole for the following reason. For any R > 0, consider the following two subsets of N (D): where C > 0 and || · || are the constant and the norm on N (D) ⊗ Z R appeared in the support property of stability conditions, see Definition 2.12.1. By the support property, we have S R is a finite set, which implies that the minimum of in the definition of systole must be attained by some σ -semistable object E. In fact, the systole must be attained by some σ -stable object: if E is a σ -semistable object and not a σ -stable object, then it is clear that the absolute value of the central charge of any non-trivial σ -stable factor of E is strictly less than |Z σ (E)|.

Remark 2.6
In the following The definition of categorical volume vol(σ ), which is also inspired by Conjecture 2.3, will be discussed in the next subsection. Now we study how the categorical systole changes under the natural group actions on Stab(D).
Proof The first three statements follow easily from the definition. To prove the last statement, we use a similar idea in the proof of [13,Theorem 4.3]. One can verify by direct computation holds for any σ and E. By Remark 2.5, there exists a σ -stable object E such that sys(σ ) = |Z σ (E)|. Since the actions by GL + (2, R) do not change the set of stable objects, E is also σ · g-stable. Therefore, is continuous on Stab(D).
Let us consider some basic examples of categorical systoles of algebraic stability conditions. In Sects. 3 and 4, we will be dealing with non-algebraic stability conditions on derived categories of coherent sheaves.
Therefore, in order to compute the categorical systoles of stability conditions in different chambers, one needs to compute the central charges of different sets of dimensional vectors.
where Quad(S, M) is the moduli space of certain meromorphic quadratic differentials on S with simple zeros. By Lemma 2.7 (2), the function sys on Stab(D) given by categorical systole descends to a function on the quotient Stab(D)/Aut(D). Under the equivalence (3), the categorical systole of a stability condition on D S,M is given by the minimum among the period integrals √ φ along saddle connections of the corresponding quadratic differential.

Categorical volumes
We recall the notion of categorical volumes of Bridgeland stability conditions introduced in [10]. It is the categorical analogue of the holomorphic volume Y ∧ of a compact Calabi-Yau manifold Y with holomorphic top form .
Let Y be a compact Calabi-Yau manifold of dimension n, and {A i } be a basis of the torsion-free part of H n (X , Z). Then one can rewrite the holomorphic volume of Y as −1 is the inverse matrix of the Euler pairings.
One can easily check that the above definition is independent of the choice of the basis {E i }: Suppose {F j } is another basis of N (D), where A is the unimodular matrix that relates the bases {E i } and {F j }, i.e. F j = k A jk E k . Denote the Gram matrices (χ(E i , E j )) and (χ(F i , F j )) by χ E and χ F , respectively. Then we have χ F = Aχ E A T , and therefore It is important to note that unlike the categorical systoles, the categorical volume of a stability condition σ = (Z, P) depends only on its central charge. The proof of the following lemma is straightforward. Now we recall some computations of categorical volumes in [10] which will be used in the later sections.

Example 2.13 (Elliptic curves) Let
Coh(E) be the derived category of coherent sheaves on an elliptic curve E. Let σ = (Z, P) be a stability condition on D with central charge where β ∈ R and ω > 0. Choose {O x , O E } as a basis of the numerical Grothendieck group N (D), where O x is a skyscraper sheaf and O E is the structure sheaf. Then the categorical volume of σ is Remark 2.14 Let C be a curve of genus g ≥ 1. Then Stab(D b Coh(C)) = C × H, and the central charge of any stability condition is of the form up to the free C-action (see [4,22]). However, the categorical volume vol(σ ) is not given by 2ω unless g = 1, due to the fact that the Euler pairing on

as a basis of N (D), then the Gram matrix of Euler pairings is
On the other hand, there exist stability conditions on D b Coh(P 1 ) that are not of this form (non-geometric). It is possible for such stability conditions to have zero categorical volume.
Note that the same computation shows that i, For any g = (T , f ) ∈ GL + (2, R), one can compute the categorical volume of the stability condition σ · g following the same idea in the proof of Lemma 2.7 (4). Write We use χ i, j = χ j,i in the last step, which follows from the fact that D is a 2-Calabi-Yau category therefore its Euler pairing is symmetric.
We should note that although the categorical volume can be defined for stability conditions on any triangulated category D, its geometric meaning is not clear unless D comes from compact Calabi-Yau geometry. Below is an example of a Calabi-Yau triangulated category for which the categorical volume vanishes for some stability conditions (on the wall of marginal stability conditions). (3-Calabi-Yau category of the A 2 -quiver) Let D be the 3-Calabi-Yau category constructed from the Ginzburg 3-Calabi-Yau dg-algebra associated to the A 2 -quiver [11,17]. The numerical Grothendieck group N (D) is generated by two spherical objects S 1 , S 2 .

Example 2.16
Let σ = (Z, P) be a stability condition on D with z 1 = Z(S 1 ) and z 2 = Z(S 2 ). Then its categorical volume is vol(σ ) = |z 1 z 2 − z 2 z 1 | = 2 |Im(z 1 z 2 )| , which vanishes if z 1 z 2 ∈ R. This can happen if z 1 and z 2 are of the same phase, i.e., the stability condition σ sits on a wall in Stab(D).

Theorem 3.1 Let D = D b Coh(E) be the derived category of an elliptic curve E. Then
holds for any σ ∈ Stab(D).
One can consider this inequality as the mirror of Loewner's torus systolic inequality in the introduction. We refer to Sect. 5 for more discussions related to mirror symmetry.
Proof By [4, Theorem 9.1], the GL + (2; R)-action on the space of Bridgeland stability conditions Stab(D) of an elliptic curve is free and transitive. Therefore By Lemma 2.7 and Lemma 2.12, the systolic ratio is invariant under the free C-action on the space of stability conditions. Hence we only need to compute the ratio on the quotient space Stab(D)/C ∼ = H. The quotient space Stab(D)/C ∼ = H can be parametrized by the normalized stability conditions as follows. Let τ = β + iω ∈ H where β ∈ R and ω > 0. The associated normalized stability condition σ τ is given by: • For 0 < φ ≤ 1, the (semi)stable objects P τ (φ) are the slope-(semi)stable coherent sheaves whose central charges lie in the ray R >0 · e iπφ . • For other φ ∈ R, define P τ (φ) by the property P τ (φ + 1) = P τ (φ) [1].
Note that there is no wall-crossing phenomenon in the elliptic curve case, i.e., the set of all Bridgeland (semi)stable objects is the same for any stability condition. This makes the computation of categorical systole easier.
To compute the systole of σ τ , by Lemma 2.7 (1), we need to know the central charges of all the stable objects of σ τ , which are the slope-stable coherent sheaves. Recall that if F is a slope-stable coherent sheaf on E, then it is either a vector bundle or a torsion sheaf. The slope-stable vector bundles on an elliptic curve E are well-understood, see for instance [2,26]. In particular, we have the following facts: Hence the categorical systole is sys(σ τ ) = min 1, | − d + τr |: (d, r ) = 1 and r > 0 where λ 1 (L τ ) denotes the least length of a nonzero element in the lattice L τ = 1, τ .
On the other hand, the categorical volume of σ τ has been computed in Example 2.13, which is equals to 2ω. Thus Note that ω is the area of the parallelogram spanned by 1 and τ . Hence the quantity sup τ ∈H λ 1 (L λ ) 2 /ω is the so-called Hermite constant γ 2 of lattices in R 2 . It is classically known that the Hermite constant is given by (see for instance [8]). This concludes the proof.

Systolic inequalities for K3 surfaces
This section is devoted to prove the following main results of the present article.

Theorem 4.1 Let X be a complex projective K3 surface. Then
Here ρ and disc denote the rank and the discriminant of the Néron-Severi group NS(X ), respectively. holds for any σ ∈ Stab † D b Coh(X ) .

Reduction to lattice-theoretic problems
We start with recalling some standard notations. Let X be a smooth complex projective K3 surface and D = D b Coh(X ) be its derived category. Sending an object E ∈ D to its Mukai vector v(E) = ch(E) √ td(X ) identifies the numerical Grothendieck group of D with the lattice The Mukai pairing on N (D) is given by   ((r 1 , D 1 , s 1 ), (r 2 , D 2 , s 2    The following proposition allows us to compute the categorical systole of a stability condition on D using only its central charge. Proof Let v = mv 0 ∈ N (D) be a Mukai vector, where m ∈ Z >0 and v 0 is primitive. A result of Bayer and Macrì [3, Theorem 6.8], which is based on a previous result of Toda [31], says that if v 2 0 ≥ −2, then there exists a σ -semistable object with Mukai vector v for any σ ∈ Stab † (D). Hence On the other hand, for any stable object E, Hence This concludes the proof.
We can now reduce the categorical systolic inequality to a lattice-theoretic problem. To prove Theorem 4.1 and 4.2, one needs to find an upper bound of the systolic ratio sys(σ ) 2 vol(σ ) for all σ ∈ Stab † (D). Since the systolic ratio is a continuous function and is invariant under the actions of autoequivalences, by Theorem 4.3 (3), it suffices to find an upper bound of the systolic ratio on U (D).
By Theorem 4.3 (2), any element in U (D) can be written as σ · g for some σ ∈ V (D) and g ∈ GL + (2, R). By Lemma 2.7 (4) and the computations in Example 2.15, we have since |t 2 | < 1 (recall the notations in Lemma 2.7). Therefore, it is enough to find an upper bound of the systolic ratio on V (D). By Theorem 4.3 (1), the central charges of stability conditions in V (D) are of the form where β, ω ∈ NS(X ) ⊗ R and ω 2 > 0. By Example 2.15, the volume of the stability conditions of this form is 2ω 2 . On the other hand, the categorical systole can be computed by Proposition 4.4 given the central charge: Hence, Theorem 4.1 and 4.2 can be obtained by proving the following lattice-theoretic statements.  |s + 2n(β + iω)d + n(β + iω) 2 r | 2 4nω 2 < n + 1; (b) nd 2 − rs ≥ −1.
We will prove these two propositions in the next two subsections.

Systolic inequality for K3 surfaces of Picard rank one
We prove Proposition 4.6 in this subsection. Note that the method in this proof does not work for K3 surfaces with higher Picard rank, due to the indefiniteness of the intersection pairing on NS(X ) for ρ(X ) > 1.
In order to find such triple (r , d, s) in Proposition 4.6 for small ω, we need the following technical lemma.

Lemma 4.7
For any real number β and any 0 < ω < 1 √ n , there exists integers (r , d, s) such that: Consider the real numbers 2nβd j + nβ 2 j 1≤ j≤l modulo 1. There is at least a pair (2nβd j + nβ 2 j, 2nβd k + nβ 2 k) has distance less than or equals to 1/l modulo 1. Say j > k without loss of generality. We choose r = j − k, d = d j − d k , and choose s to be the integer closest to −2nβd − nβ 2 r . Then Let = s + 2nβd + nβ 2 r . Then We have Hence −1 < nd 2 − sr < n + 1. Since it is an integer, thus 0 ≤ nd 2 − sr ≤ n.
We can now prove Proposition 4.6.

Proof of Proposition 4.6
If ω ≥ 1 √ n , one can simply take the class of skyscraper sheaves (r , d, s) = (0, 0, 1) and check that it satisfies the required conditions. If ω < 1 √ n , we choose (r , d, s) as in Lemma 4.7. Then it satisfies sr < nd 2 + 1 and It is not hard to show that the notions of spherical systole and categorical systole coincide for stability conditions on elliptic curves. However, this is not true for the derived categories of K3 surfaces. The following proposition shows that the systolic inequality does not hold for spherical systole on K3 surfaces.
Here we use the fact that the Mukai vector of a spherical object S satisfies v(S) 2 but we only use the Mukai vectors that satisfy 0 ≤ v 2 ≤ 2n to prove the systolic inequality (c.f. Lemma 4.7).

Systolic inequality for general K3 surfaces
We prove Proposition 4.5 in this subsection. The proof is based on a classical result on the existence of integer points by Minkowski.
Theorem 4.11 (Minkowski) Every convex set in R n which is symmetric with respect to the origin and has volume greater than 2 n contains a non-zero integer point.
Proof of Proposition 4. 5 We fix an identification NS(X ) ∼ = Z ρ . For any ω ∈ NS(X ) ⊗ R with ω 2 > 0, the intersection pairing restricts on ω ⊥ ⊂ R ρ is negative definite by Hodge index theorem. Choose where C is any number larger than (ρ+2)!
Consider the following convex set in R ρ+2 which is symmetric with respect to the origin: One can check that the volume of C is greater than 2 ρ+2 . Therefore to prove Proposition 4.5, it suffices to show that any vector in C satisfies the following conditions: Observe that the vectors v 1 , v 2 , . . . , v ρ+2 satisfy all three conditions. Since the first two conditions are linear in (r , D, s) ∈ R ρ+2 , hence are satisfied by all the vectors in C. Then This concludes the proof of Proposition 4.5 by Minkowski's theorem.

Applications in symplectic geometry
Recall the two questions we proposed in the introduction. In the present article, we give an affirmative answer to Question 5.2 for any complex projective K3 surface X , and find an explicit systolic constant C which depends only on the rank and discriminant of NS(X ). Now we discuss how this result can be used to answer Question 5.1, via the homological mirror symmetry conjecture proposed by Kontsevich.

Conjecture 5.3 [18]
For any Calabi-Yau manifold with a symplectic structure (Y , ω) , there exists a Calabi-Yau manifold with a complex structure (X , ) such that there is an equivalence between triangulated categories The conjecture has been proved in several cases, see for instance [27][28][29]. In particular, for any K3 surface Y and a symplectic form ω on Y , it is expected that there exists a K3 surface X with a complex structure such that the above equivalence of categories holds. Proof Conjecture 2.3 holds for D b Fuk(Y , ω) implies that for any holomorhpic 2-form Y on Y , there is an associated Bridgeland stability condition σ Y on the derived Fukaya category of (Y , ω) such that sys(Y , ω, Y ) = sys(σ Y ) and Y Y ∧ Y = vol(σ Y ).
Assuming the validity of Conjecture 5.3 for K3 surface (Y , ω), then there exists a K3 surface with a complex structure (X , ) such that Therefore σ Y induces a Bridgeland stability condition on D b Coh(X , ). By Theorem 1.7 and the assumption that Stab(D b Coh(X )) is connected, we have where C = ((ρ+2)!) 2 |disc NS(X )| 2 ρ + 4. Hence, the inequality holds for any holomorphic 2-form Y on Y .

Systolic inequality for Calabi-Yau threefolds
The existence of Bridgeland stability conditions on quintic Calabi-Yau threefolds is proved by Li recently [21]. It would be interesting to investigate whether the categorical systolic inequality sys(σ ) 2 ≤ C · vol(σ ) continues to hold for Calabi-Yau threefolds. Note that the categorical volumes of geometric stability conditions on quintic Calabi-Yau threefolds near the "large volume limit" were computed in [10,Section 4.4]. Hence the main difficulty lies in determining the categorical systoles. It would be nice if properties similar to Proposition 4.4 hold for Calabi-Yau threefolds.

Categorical systole as topological Morse function
It is proved by Akrout [1] that the systole of Riemann surfaces is a topological Morse function on the Teichmüller space. In fact, Akrout shows that a "generalized systolic function" defined locally as the minimum of a finite number of functions with positive definite Hessians is a topological Morse function, and then apply a result of Wolpert [32] that length functions have positive definite Hessians with respect to the Weil-Petersson metric on the Teichmüller space.
Motivated by the correspondence between flat surfaces and stability conditions, it would be interesting to show that the categorical systole is a topological Morse function on the space of Bridgeland stability conditions, and then deduce some topological properties of the quotient space Stab(D)/Aut(D). Since there also is a categorical analogue of Weil-Petersson metric on the space of stability conditions studied in [10], one might be able to follow the same approach as Akrout's proof. We hope to come back to this question in the future.