A note on Bogomolov-Gieseker type inequality for Calabi-Yau 3-folds

The conjectural Bogomolov-Gieseker (BG) type inequality for tilt semistable objects on projective 3-folds was proposed by Bayer, Macri and the author. In this note, we prove our conjecture for slope stable sheaves with the smallest first Chern class on certain Calabi-Yau 3-folds, e.g. quintic 3-folds.

1. Introduction 1.1. Motivation and result. Let X be a smooth projective 3-fold over C. Given an element with ω ample, the heart of a bounded t-structure B B,ω ⊂ D b Coh(X) was constructed in [3], following the construction of Bridgeland's stability conditions on projective surfaces [5], [1]. The notion of tilt stability on B B,ω was introduced in [3], and a conjectural Bogomolov-Gieseker (BG) type inequality among Chern characters of tilt semistable objects in B B,ω was proposed in [3,Conjecture 1.3.1]. Our conjecture in [3] turned out to imply several very important results: construction of Bridgeland stability on projective 3-folds [3], Fujita's conjecture in birational geometry [2], and Ooguri-Strominger-Vafa's conjecture in string theory [9]. In this note, we report a partial progress toward the conjectural BG type inequality in [3].
When B = 0, the first Chern class on the heart B 0,ω is always nonnegative, and plays a role of the rank on the category of coherent sheaves. A hopeful approach toward the proof of the main conjecture in [3] is to use the induction argument on the first Chern classes of tilt semistable objects, as in the proof of BG type inequality without ch 3 . (cf. [3,Theorem 7.3].) As a first step for this induction argument, the conjectural BG type inequality should be solved when the tilt semistable object has the smallest first Chern class. In this case, the required result is formulated in the following conjecture for slope stable sheaves (cf. [ The above conjecture was studied in [3,Example 7.2.4] for rank one torsion free sheaves. In this case, the inequality (1) is reduced to Castelnuovo type inequality for low degree curves in X. On the other hand, the higher rank case was not studied in [3]. The purpose of this article is to show that, when X is a certain Calabi-Yau 3-fold, the inequality (1) is reduced to Castelnuovo type inequality even in the higher rank case. The main result is as follows: Theorem 1.2. Let X be a smooth projective Calabi-Yau 3-fold such that Pic(X) is generated by O X (H) for an ample divisor H in X. Suppose that the following inequalities hold: for any one dimensional subscheme C ⊂ X with C ·H < H 3 /2. Then X satisfies Conjecture 1.1. Furthermore the inequality (1) is an equality only when E = O X (H).
As we discussed in [3, Example 7.2.4], a typical (and important) example satisfying the assumption of Theorem 1.2 is a quintic 3-fold in P 4 . Therefore we obtain the following corollary: Corollary 1.3. Let X ⊂ P 4 be a smooth quintic 3-fold. Then X satisfies Conjecture 1.1.
In the case of quintic 3-folds, the conditions c 1 (E) = [H], ch 2 (E)H > 0 and the Bogomolov-Gieseker inequality [4], [6] restrict the rank of E up to five. So a priori, the sheaf E could be rank(E) ≥ 2. On the other hand, we do not know any example of such a sheaf E with rank(E) ≥ 2. (cf. Remark 2.5.) The result of Corollary 1.3 means that ch 3 (E) should obey the desired inequality (1), if such a sheaf E exists.
In general, the third Chern character ch 3 (E) is known to be bounded by a certain polynomial of ch 0 (E), ch 1 (E) and ch 2 (E), see [8]. However 1 The statement of [3, Conjecture 7.2.3] was more general than Conjecture 1.1 and the formulation is slightly different. When Pic(X) is generated by one element, they are obviously equivalent. the evaluation in [8] is not strict to show the inequality (1). Although the hypersurface restriction of E plays an important role in [8], we do not take the hypersurface restriction. Instead we take the universal extension and the classical Bogomolov-Gieseker inequality to evaluate the dimensions of cohomology groups. As far as the author knows, such a method is not seen in literatures.

Acknowledgement. This work is supported by World Premier
International Research Center Initiative (WPI initiative), MEXT, Japan. This work is also supported by Grant-in Aid for Scientific Research grant (22684002), and partly (S-19104002), from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

Notation and convention.
In this note, all the varieties are defined over C.
We say X is a Calabi-Yau 3-fold if dim X = 3, its canonical line bundle is trivial and h 1 (O X ) = 0. For an ample divisor H in a 3-fold X and a torsion free sheaf E on X, its slope is denoted by The notion of slope stability is defined in the usual way. (cf. [7].) For a subscheme Z ⊂ X, the defining ideal sheaf of Z is denoted by I Z .
2. Proof of Theorem 1.2 2.1. Some lemmas. The key ingredient for the proof of Theorem 1.2 is the following two lemmas, which may be well-known. For the lack of references, we give the proofs.
the sheaf E ′ is also slope stable.
Proof. We prove the assertion by the induction on ext 1 (E, O X ). When ext 1 (E, O X ) = 0, then the assertion is obvious.
Suppose that ext 1 (E, O X ) > 0, and take a non-zero element a ∈ Ext 1 (E, O X ). The element a corresponds to the extension, We show that E a is slope stable. Suppose by contradiction that E a is not slope stable. Then there is a saturated subsheaf F ⊂ E a such that F is slope stable and If we write c 1 (F ) = k[H], then k ≥ 1, hence Hom(F, O X ) = 0. It follows that the composition is non-zero, which implies µ H (F ) ≤ µ H (E). Combined with (6), we obtain the inequality The above inequality immediately implies k = 1 and r(F ) = r(E).
Then µ H (F ) = µ H (E), and since F and E are slope stable with the same slope, the non-zero morphism (7) is an isomorphism. However this contradicts to that the sequence (5) is non-split. Let V a be the C-vector space Ext 1 (E a , O X ) ∨ and take the universal extension, Applying Hom(−, O X ) to the sequence (5), we see that Hence E ′ a is slope stable by the assumption of the induction. On the other hand, composing the sequence (5) with (8), we obtain the exact sequence It is easy to see that the above sequence is identified with the sequence (4), hence E ′ ∼ = E ′ a is slope stable.

Lemma 2.2. In the situation of Lemma 2.1, suppose that r(E) ≥ 2
and there is a non-zero element s ∈ H 0 (X, E). Then for the associated exact sequence Proof. We first show that F is torsion free. If F has a torsion, there is an exact sequence where T is a non-zero torsion sheaf and A ⊂ E is a rank one torsion free sheaf. If dim Supp(T ) = 2, then c 1 (A) = k[H] with k ≥ 1, which contradicts to that E is slope stable. Therefore dim Supp(T ) ≤ 1, hence Therefore the sequence (9) splits, which contradicts to that A is torsion free.
Next suppose that F is not slope stable. Then there is a slope stable sheaf G and a surjection F ։ G satisfying Also since there is a surjection E ։ F ։ G and E, G are slope stable, It is immediate to see that there is no solution (k, r(G)) satisfying the above inequality and r(G) < r(E) − 1. Hence F is slope stable.
As a corollary of Lemma 2.2, we have the following:

Corollary 2.3. In the situation of Lemma 2.1, there is an exact sequence of the form
such that F is either a rank one torsion free sheaf or a slope stable sheaf with r(F ) ≥ 2 and h 0 (F ) = 0.
Proof. We show the assertion by the induction of θ(E) defined by The assertion is obvious when θ(E) = 0. Suppose that θ(E) > 0, i.e. h 0 (E) = 0 and r(E) ≥ 2. Then there is a non-zero element s ∈ H 0 (X, E). If we take the exact sequence then F s is slope stable by Lemma 2.2. By applying Hom(O X , −) to the sequence (11), we see h 0 (F s ) = h 0 (E) − 1. Hence we have θ(F s ) = θ(E) − 1, and by the assumption of the induction, there is an exact sequence such that F is a rank one torsion free sheaf or a slope stable sheaf with r(F ) ≥ 2 and h 0 (F ) = 0. The desired exact sequence (10) is obtained by combining the sequence (12) with (11).

Proof of Theorem 1.2.
Proof. Let X be as in the statement of Theorem 1.2, and E a slope stable sheaf on X with c 1 (E) = [H] and ch 2 (E)H > 0. By Corollary 2.3, there is an exact sequence of the form such that either F is a rank one torsion free sheaf or a slope stable sheaf with r(F ) ≥ 2 and h 0 (F ) = 0. Note that in the first case, we have F ∼ = O X (H) ⊗ I Z for a subscheme Z ⊂ X with dim Z ≤ 1. We evaluate ch 3 (E) = ch 3 (F ) by dividing into the following three cases: In this case, we have by the Serre duality, which is zero by the cohomology exact sequence associated to the sequence and the Kodaira vanishing h 2 (O X (H)) = 0. Hence the sequence (13) splits if m > 0, which contradicts to the slope stability of E. Therefore E ∼ = O X (H) ⊗ I Z , and The above equalities imply the inequality (1), and the equality holds only when Z = ∅.
In this case, ch 2 (E)H = ch 2 (F )H > 0 is equivalent to Applying the assumption (3), we have Therefore the inequality (1) holds.