On stability conditions for the quintic threefold

We study the Clifford type inequality for a particular type of curves $C_{2,2,5}$, which are contained in smooth quintic threefolds. This allows us to prove some stronger Bogomolov-Gieseker type inequalities for Chern characters of stable sheaves and tilt-stable objects on smooth quintic threefolds. Employing the previous framework by Bayer, Bertram, Macr\`i, Stellari and Toda, we construct an open subset of stability conditions on every smooth quintic threefold in $\mathbf{P}^4_{\mathbb C}$.


INTRODUCTION
The notion of stability conditions on a triangulated category is introduced by Bridgeland in [Bri07]. The existence of stability conditions on three-dimensional projective varieties, and more specifically on Calabi-Yau threefolds, is often considered as one of the biggest open problem in the theory of Bridgeland stability conditions in recent years. In series work of [BMT14,BBMT14,BMS16], the authors propose a general approach towards the constructions of geometric stability conditions on a smooth projective threefold. The construction involves the notion of tilt-stability for two-term complexes, and the existence of geometric stability conditions relies on a conjectural Bogomolov-Gieseker type inequality for the third Chern character of tilt-stable objects.
Stability conditions are only known to exist on few families of smooth projective threefolds: Fano threefolds [Mac14,Sch13,Li15,Piy16,BMSZ17], Abelian threefolds [MP13a,MP13b,BMS16] and Kummer type threefolds [BMS16]. The smooth quintic threefolds will be the first example of projective Calabi-Yau threefolds with a trivial fundamental group. One need to be cautious that the original conjectural Bogomolov-Gieseker type inequality in [BMT14] does not hold for all threefolds, counterexamples for the blowup at a point of another threefold has been constructed in [Sch17,MSD17]. However, due to the flexibility of the construction in [BMT14] as well as the work [PT15], modified Bogomolov-Gieseker type inequality will still imply the existence of stability conditions on such threefolds.
In a special case that when H 2 ch 1 (F ) H 3 rk(F ) = − 1 2 , we have ∆(E)H ≥ 1.25 rk(E) 2 , which is a slightly weaker inequality than that in [Tod17, Conjecture 1.2]. In particular, it implies the rank 2 case as that in [Tod17, Proposition 1.3].
Theorem 1.1 implies [BMS16,Conjecture 4.1] for smooth quintic threefolds with a little constrain on the parameters (α, β), for which we will review in the next few paragraphs.
Employing the framework in [BMS16,BMT14,PT15], Theorem 1.2 allows us to construct a family of Bridgeland stability conditions on the bounded derived category of coherent sheaves on each smooth quintic threefold. To give the accurate statement, we introduce some notions from [BMT14,BBMT14,BMS16] and briefly summarize the construction of stability conditions on a quintic threefold.
Stability conditions on smooth quintic threefolds: Let (X, H) be a smooth quintic threefold with H = [O X (1)], let D b (X) be the bounded derived category of coherent sheaves on X. As shown in [Bri07,Proposition 5.3], a stability condition on D b (X) is equivalently determined by a pair σ = (Z, A), where the central charge Z : K 0 (A) → C is a group homomorphism and A ⊂ D b (X) is the heart of a bounded t-structure, which have to satisfy the following three properties.
(a) For any non-zero object E ∈ A, its central charge Z([E]) ∈ R >0 · e (0,1]πi . This allows us to define a notion of slope-stability on A via the slope function , when H 2 ch βH 1 (E) > 0; +∞, when H 2 ch βH 1 (E) = 0. (1) The explicit formulas of twisted Chern characters ch βH i are given at the beginning of Section 2. The heart A α,β, This family is a slice of the GL   [Bay16] which reproves the Brill-Noether generality of certain curves on K3 surfaces as that in [Laz86]. The estimation for hom(O X , E) necessarily relies on a stronger Bogomolov-Gieseker type inequality for the second Chern character of slope stable objects, which is the statement of Theorem 5.5. The argument in 3 relies on two techniques: the deformation of stability conditions and Feyzbakhsh's restriction lemma. A similar deformation argument has been used in [Li15] for the case of Fano threefolds with index one. The restriction lemma first appears in [Fey16], where the author shows the stability of vector bundles on curves restricted from a K3 surface. More details about the restriction technique via stability conditions appear in Feyzbakhsh's thesis. The argument 3 can produce more Bogomolov-Gieseker type inequalities for the first two Chern characters for several other varieties. Some results focused on this direction will appear soon in [Li19]. 4 Proposition 4.1 is the Clifford type bound for the dimension of global sections of stable vector bundles on curves C 2,2,5 , the complete intersection of two quadratics and a quintic hypersurface in P 4 . As a topic of its own interest, several general results on the Clifford type bound for curves can be found in [AFO14,LN15a,LN15b,LN17,Mer02]. It is pity that none of the results mentioned above fit in our situation since we need the sharp bounds at some critical slopes µ = 5, 10, 30 and 35. Based on the idea in [Fey17], together with Feyzbakhsh, we develop our own methods to estimate the Clifford type bound for curves supported on K3 and Fano surfaces via stability conditions in [FL18]. Especially for this case, we think C 2,2,5 as a curve on a degree four del Pezzo surface. More introductions about the technical details in 4 can be found in the two papers mentioned above. We organize the paper slightly different from the logic flow. Section 2 is to fix some notations and to collect some lemmas and tools that will be useful in every other section. In section 3, we assume the result in Theorem 5.5 and directly prove our main Theorem 2.8. We make this arrangement since the arguments in this part are more well-established, also we would like to convince the reader that a stronger Bogomolov-Gieseker type inequality for the second Chern character of slope stable sheaves will imply Bogomolov-Gieseker type inequality for the third Chern character of tilt-stable complexes at this early stage. Section 4 is devoted to proving the Clifford type bound for the dimension of global sections of a stable vector bundle on the curve C 2,2,5 . This section involves a certain amount of computations. As for the convenience of the readers, there is no harm to skip these details first. Section 5 is to proof the stronger Bogomolov-Gieseker type inequality for the surfaces S 2,5 based on the inequality in Proposition 4.1.
Acknowledgement: The author is a Leverhulme Early Career Fellow at the University of Warwick and would like to acknowledge the Leverhulme Trust for the support. The work was initiated when the author was a postdoc at the University of Edinburgh and was supported by the ERC starting grant 'WallXBirGeom' 337039. I am grateful to Arend Bayer, Soheyla Feyzbakhsh, Emanuele Macrì, Laura Pertusi, Benjamin Schmidt, Junliang Shen, Paolo Stellari and Xiaolei Zhao for many useful discussions on this topic. The main breakthrough of this project is done during my visit at BICMR in July 2018. I would like to thank Zhiyu Tian, Chenyang Xu and Qizheng Yin for their hospitality.
2. BACKGROUND: TILT-STABILITY CONDITION AND WALL-CROSSING 2.1. Stability condition: notations and conventions. In this section, we review the notion of stability and tilt-stability for smooth varieties introduced in [Bri07, BMT14,PT15]. We then recall the conjectural Bogomolov-Gieseker type inequality for tilt-stable complexes proposed there.
Let X be a smooth projective complex variety and H ∈ N S(X) R be a real ample divisor class. Let the dimension of X be n, in this paper, n will always be 2 or 3. For an arbitrary divisor class B ∈ N S(X) R , we will always denote the twisted Chern characters as follows: In this paper, we are mainly interested in smooth quintic threefold whose N S(X) R is of rank 1, we will always assume B = βH for some β ∈ R. The µ H -slope of a coherent sheaf E on X is defined as H n ch 0 (E) , when ch 0 (E) = 0; +∞, when ch 0 (E) = 0. Each coherent sheaf E admits a unique Harder-Narasimhan filtration: There exists torsion pairs (T β,H , F β,H ) in Coh(X) defined as follows: Definition 2.2. We let Coh β,H (X) ⊂ D b (X) be the extension-closure By the general theory on tilting heart in [HRS96], Coh β,H (X) is the heart of a t-structure in D b (X). Given α ∈ R, we may define the tilt-slope function for objects in Coh β,H (X) as follows: for an object E ∈ Coh β,H (X), its tilt-slope function , when H n−1 ch βH 1 (E) > 0; +∞, when H n−1 ch βH 1 (E) = 0. Definition 2.3. An object E ∈ Coh β,H (X) is called ν α,β,H -tilt slope (semi)stable if for any nontrivial subobject F ֒→ E in Coh β,H (X), we have The tilt slope stability also admits Harder-Narasimhan property when α > β 2 2 . For an object E ∈ Coh H,β (X) we may write ν + α,β,H (E) and ν − α,β,H (E) for the maximum and minimum slopes of its semistable factors respectively.
Definition 2.5. Let E be an object in D b (X), we define its H-discriminant as Theorem 2.6 (Bogomolov Inequality, [BMT14, Theorem 7.3.1], [PT15, Proposition 2.21]). Let X be a smooth projective variety, and H ∈ N S(X) R an ample class. Assume that E is ν α,β,H -tilt semistable for some α > 1 2 β 2 , then∆ H (E) ≥ 0. The main goal of this paper is on the following conjectural Bogomolov-Gieseker inequality for ν α,β,H -tilt semistable objects: Conjecture 2.7 (Conjecture 4.1 in [BMS16]). Let X be a smooth projective threefold, and H ∈ N S(X) R an ample class. Assume that E is ν α,β,H -tilt semistable for some α > 1 2 β 2 , then In this paper, we will prove this conjecture for smooth quintic threefolds with a little assumption on α.
Theorem 2.8. Let X be a smooth projective quintic threefold, and , then the inequality (2) holds.
2.2. Recollection of lemmas. Let X be a smooth projective variety and H ∈ N S(X) R be a real ample divisor class. For an object Let α, β ∈ R be the parameters for tilt-slope functions, unless mentioned otherwise, we will always assume α > 1 2 β 2 . Lemma 2.9. Let E ∈ Coh β 0 ,H (X) be a ν α 0 ,β 0 ,H -tilt stable object for some α 0 > 1 2 β 2 0 , then we have the following properties.
(a) (Openness) There exists an open set of neighborhood U of (α 0 , β 0 ) such that for any The statement also holds for semistable case. Moreover, when X is a threefold, More precisely, the requirements on E and F are as follows: bothv H (E) andv H (F ) are not zero and the determinant This equals the right hand side since the zero determinant implies: The following lemma from [BMS16] will be very useful in the technique of deforming tiltstabilities. We list it here for the convenience of readers.

The equality holds only whenv
Definition 2.11. We call an object E Brill-Noether stable if there exists an open subset We call an object E Brill-Noether semistable if there exists δ > 0 such that E is ν α,0,H -tilt semistable for every 0 < α < δ.
For an object E ∈ Coh 0,H (X), we denote its Brill-Noether slope by By Lemma 2.9, an object E with H n−2 ch 2 (E) = 0 is Brill-Noether stable if and only if it is ν α,β,H -tilt stable for some (α, β) proportional to p H (E). The Brill-Noether semistability of E implies that E is ν α,β,H -tilt semistable for some (α, β) proportional to p H (E).
, O X )) * be a subspace, then the object is in Coh 0,H (X) and Brill-Noether semistable.
Proof. We prove the case when ν BN (E) > 0, the other case can be proved in a similar way. Note thatẼ is the canonical extension In the case that ν BN (E) = +∞, for any α > 0, both E and O X [1] ⊗ W are ν α,0,H -tilt semistable with the same slope +∞. Any extended object from them, especiallyẼ, is also ν α,0,H -tilt semistable with slope +∞.

PROOF FOR THE MAIN RESULT
The goal of this section is to prove the inequality in Theorem 2.8 with the assumption of Theorem 5.5. Following the idea in [BMS16, Section 5], we first reduce the inequality for every tilt semistable objects to Brill-Noether stable objects.
Consider the wall W through (α, β) and p H (E): is to the right of the dashed lines.

CLIFFORD TYPE INEQUALITY FOR CURVES C 2,2,5
The generalized Clifford index theorem for curves, [BPGN95, Theorem 2.1] states that for any semistable vector bundle F over a smooth curve with rank r and slope µ ∈ [0, g], the following bound holds: The main purpose of this section is to set up the following stronger Clifford type inequality for the curve C 2,2,5 , which is the complete intersection of two quadratic hypersurfaces and a quintic hypersurface in P 4 C . Proposition 4.1. Let F be a semistable vector bundle on a smooth curve C 2,2,5 with rank r and slope µ ∈ (0, 10] ∪ [30, 40], then we have the following bounds for the dimension of global sections of h 0 (F ): . Bounds for h 0 (F )/r when µ ∈ (0, 5).
It is enough to prove the statement for stable vector bundles. Let S 2,2 be a smooth complete intersection of two quadratic hyper-surfaces such that S 2,2 contains C 2,2,5 in P 4 C . We denote the inclusion map by ι : C 2,2,5 ֒→ S 2,2 . In this section, we write H for [O S 2,2 (1)] and only use stability conditions on S 2,2 with polarization H. We will always consider the dimension of global sections on ι * F in D b (S 2,2 ) instead of F . The following statement is standard: Lemma 4.3. Let F be a stable vector bundle on C 2,2,5 , then ι * F is ν α,0,H -tilt stable for α ≫ 0.
Following the strategy in [Fey17] and [FL18], we will compute h 0 (ι * F ) by considering the Harder-Narasimhan factors of ι * F with respect to ν BN .
Proof. The inequality for Chern characters of µ H -slope stable objects with H ch 1 (E) H 2 rk(E) / ∈ Z is by computing The stability condition is then a standard construction as that in [Bri08] or the framework in [PT15, Section 2].
The following lemma explains that we can estimate the dimension of global sections for each Brill-Noether semistable factor.
We finish the claim for all cases.
If s = 0, then H −1 (F 1 ) = 0 as it is torsion free. Since F 2 and ι * F have the same ν α,0,H slope, we must have ch(ι * F ) = k ch(F 2 ) for some real number k > 0, this will violate the stability assumption on F . Thus, we may assume s = 0.
As for the ν BN (E m /E m−1 ), one may use the same argument and reduce it to the computation of Γ(β 1 ) β 1 .
Lemma 4.11. Let O = (0, 0) be the origin, let P = (x p , y p ) and Q = (x q , y q ) be two points on H such that xp yp < xq yq and y p > y q . Consider all collections of points P 0 = O, P 1 , . . . , P n = P in the triangle OQP such that P 0 P 1 . . . P n P 0 forms a convex polygon, then the sum can achieve its maximum when either n = 1 or 2. In addition, when n = 2, the point P 1 = (x 1 , y 1 ) can be chosen on the line segment OQ (QP , respectively) unless Proof. Consider the following toy model on the left in Figure 5: y c > y b > y a and AC//A ′ C ′ . We allow A ′ to move alone the line segment AB (C ′ moves along BC accordingly so that AC//A ′ C ′ ). Note that the function ♣( changes linearly with respect to the length of AA ′ , it can achieve maximum when either A ′ = C ′ = B or both A ′ = A and C ′ = C. Therefore, to achieve the maximum of (18) we may remove extra P i 's when n > 2. Back to the case of the lemma when n = 2, we may always adjust the position of P 1 so that it satisfies the requirements in the statement.
Proof of Proposition 4.1. It is enough to prove the case for slope stable vector bundle F over C 2,2,5 . We consider the Harder-Narasimhan filtration for ι * F with respect to ν BN as that in Lemma 4.4: . The HN polygon for ι * E is inside the triangle OQP .
We draw the points P i := (ch 2 (F i ), H ch 1 (F i )), 1 ≤ i ≤ m on the upper half plane H. By Lemma 4.8 and the definition of the function ♣, Let P = P n = ((µ − 50)r, 20r) and Q = (x q , y q ) be points on H such that xq yq is the upper bound for ν + BN (ι * F ) and xp−xq xq−yq is the lower bound for ν − BN (ι * F ) as that in Proposition 4.9. The points O, P 1 , . . . , P, O then form the vertices of a convex polygon in the triangle OQP as that in Figure 6. Now by Lemma 4.11, we may estimate the upper bound for h 0 (F ) by choosing suitable candidate point P 1 := (x 1 , y 1 ) in the triangle OQP .
Note that y 1 as a function of µ is convex down when µ ≤ 10. Substituting y 1 ( 5 2 ) = -In the third case, the coordinate of P 1 is given in the second case of µ ∈ [2, 5 2 ). The term 1 656 y 1 in (20) is 35 20336 r and 5 1353 r when µ = 5 2 and 10 3 respectively. The term 5 492 y 1 in (21) is 5 984 r and 200 18819 r when µ = 5 2 and 10 3 respectively. Therefore, (20) is always less than the estimation in the second case.
We may consider when P 1 = Q or P 1 is on the line segment OQ such that The goal of this section is to prove the stronger Bogomolov-Gieseker type inequality for the second Chern character of slope stable sheaves on a quintic threefold. Our strategy is to first reduce this to the same inequality for a surface on the quintic threefold.
The following Feyzbakhsh's restriction lemma [Fey16] will be one of the key tools to reduce Bogomolov-Gieseker type inequality for higher dimensional varieties to surfaces. Then for a generic smooth irreducible subvariety Y ∈ |mH|, the restricted sheaf E| Y is µ H Yslope semistable on Y . Moreover, rk(E) = rk(E| Y ), H n−2 Y ch 1 (E| Y ) = mH n−1 ch 1 (E) and when n = 3, ch 2 (E| Y ) = mH ch 2 (E).
a factor F i with H ch 1 (F i ) Without loss of generality, we may assume H ch 1 (F ) As a summary, our methods are expect to be generalized to some other Calabi-Yau threefolds, meanwhile it seems that each deformation type will require much computation.