On stability conditions for the quintic threefold

We study the Clifford type inequality for a particular type of curves C2,2,5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{2,2,5}$$\end{document}, 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, Macrì, Stellari and Toda, we construct an open subset of stability conditions on every smooth quintic threefold in PC4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {P}^4_{\mathbb {C}}$$\end{document}.


Introduction
The notion of stability conditions on a triangulated category is introduced by Bridgeland [10]. 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 [4,5,7], 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 [6,18,24,29,31], Abelian threefolds [5,26,27] and Kummer type threefolds [5]. The smooth quintic threefolds will be the first example of strict Calabi-Yau threefolds that has geometric stability conditions. One need to be cautious that the original conjectural Bogomolov-Gieseker type inequality in [7] does not hold for all threefolds, counterexamples for the blowup at a point of another threefold has been constructed in [28,32]. However, due to the flexibility of the construction in [7] as well as the work [30], modified Bogomolov-Gieseker type inequality will still imply the existence of stability conditions on such threefolds.
In this paper, we prove the following Bogomolov-Gieseker type inequalities for the second Chern character of slope stable sheaves on smooth quintic threefolds: In a special case that when H 2 ch 1 (F) H 3 rk(F) = − 1 2 , we have (F)H ≥ 1.25rk(F) 2 , which is a slightly weaker inequality than that in [33,Conjecture 1.2]. In particular, it implies the rank 2 case as that in [33,Proposition 1.3].
Theorem 1.1 implies [5,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. Theorem 1.2 (Theorem 2.8) Conjecture 4.1 in [5] holds for smooth quintic threefolds when the parameters satisfy α 2 + (β − β − 1 2 ) 2 > 1 4 . Employing the framework in [5,7,30], 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 [4,5,7] 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 [10,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. (b) With respect to the slope-stability ν σ , each non-zero object E ∈ A admits a unique Harder-Narasimhan filtration: such that: each quotient F i := E i /E i−1 is μ σ -slope semistable with ν σ (F 1 ) > ν σ (F 2 ) > · · · > ν σ (F m ). We set ν + σ (E) := ν σ (F 1 ) and ν − σ (E) := ν σ (F m ). (c) (support property) There is a constant C > 0 such that for all semistable object E ∈ A, we have [E] ≤ C |Z ([E])|, where · is a fixed norm on K 0 (X ) ⊗ R.
Under the framework of [4,5,7], the heart A of the stability condition is constructed by 'double-tilting' Coh(X ). Denote μ H as the slope stability on Coh(X ). For any object E ∈ Coh(X ), let μ + H (E) (μ − H (E)) be the maximum (minimum) slope of its Harder-Narasimhan factors. The first tilting-heart Coh β,H (X ) ⊂ D b (X ) with parameter β ∈ R is the extension-closure T β,H , F H,β [1] , where Given α ∈ R >0 , 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 is defined as The explicit formulas of twisted Chern characters ch β H i are given at the beginning of Sect. 2.
The heart A α,β, The central charge on A α,β,H (X ) is defined as that in [5,Lemma 8.3]: As a corollary of [5,Conjecture 4.1] employing the framework in [4,5,7], the construction above offers us a family of stability conditions.
; and a > α 2 6 This family is a slice of the G L The mirror family of X is parameterized by the stack M K , which is called the stringy Kähler moduli space of X : Here the generator of μ 5 acts on C by the multiplication of e 2πi 5 . Based on the papers [2], [12,Remark 3.9] and [33], it is expected that there is an embedding from the stringy Kähler moduli space to the double quotient: We refer readers to [2, Section 7.1] and [33, Section 3] for more detailed discussions and predictions on the formula of centrals charge and heart structures. Under this embedding, the images of (the neighbourhoods of) three special points are of particular interests: • the large volume(radius) limit at the point ψ = ∞; • the conifold gap point at the point ψ 5 = 1; • the Gepner point the point ψ = 0.
Up to the actions by Aut(D b (X )) and C, the images of the neighborhood of the large volume limit are expected to be expressed by geometric stability conditions with predicted central charge: where β ∈ R and t > 0. Scaling the imaginary part of (2) by t, let b = 0, a = t 2 2 and α = √ 15 3 t; we get all such central charges for t > 1. In particular, the space of stability conditions constructed in Theorem 1.3 contains a neighbourhood of the large volume limit.
Up to the actions by Aut(D b (X )), the limit of central charges near the conifold gap point is expected to satisfy Z (O X ) = 0. Note that in (2), one may let β = 0 and α → 0 so that the kernel of the central charge will tend to the character of O X . In particular, the space of stability conditions constructed in Theorem 1.3 contains parts of the neighbourhood of conifold gap point.
The image of the Gepner point is expected to be represented by a stability condition that is fixed by the action (ST O X • ⊗O(H ), − 2

Organisation and Approach
The logic flow of the proof is as follows: 4  Each statement above is an inequality for characters of certain semistable objects. Each ' ⇒ ' only relies on the previous inequality but not relates to the arguments for that inequality. The argument in 1 follows the technique in [5,Section 5], it is also originated from the idea in [24,Section 2.2]. Naively speaking, by 1 we may reduce the inequality for stable objects with respect to every tilt-slope functions to a single type: the so-called 'Brill-Noether' stable objects. The mainstream of the argument in 2 is to follow the technique developed for Fano threefolds as that in [6,18,29]. However, in the Calabi-Yau threefold case, we don't have some of the Hom vanishings as that in the Fano threefolds case. Instead, we need to estimate the hom(O X , E) for Brill-Noether stable objects. The original idea for this estimation via stability conditions, as far as the author knows, first appears in [3] which reproves the Brill-Noether generality of certain curves on K3 surfaces as that in [17]. 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. In addition to Proposition 4.1, 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 [18] for the case of Fano threefolds with index one. The restriction lemma first appears in [13], 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 [19]. 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 [1,[20][21][22]25]. 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 [14], 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 [15]. 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 [14,15]. 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 Sect. 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.
2 Background: tilt-stability condition and wall-crossing

Stability condition: notations and conventions
In this section, we review the notion of stability and tilt-stability for smooth varieties introduced in [7,10,30]. 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 = β H 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: By the general theory on tilting heart in [16], 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 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.
We also write the central charge for an object E ∈ Coh β,H (X ).

Remark 2.4
The formula ν α,β,H is re-parameterized from the one in [5,Section 4]. Let the tilt-slope function in [5, Section 4] be ν α,β,H , then In particular, an object E ∈ Coh β,H (X ) is ν α,β,H -tilt (semi)stable (in the sense of [5]) if and only if ν 1 2 (α 2 +β 2 ),β,H -tilt (semi)stable. We use ν α,β,H as it is more convenient to compare the slopes of objects via pictures. The main goal of this paper is on the following conjectural Bogomolov-Gieseker inequality for ν α,β,H -tilt semistable objects: In this paper, we will prove this conjecture for smooth quintic threefolds with a little assumption on α.

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 .

then we have the following properties. (a) (Openness) There exists an open set of neighborhood U of
The statement also holds for semistable case. Moreover, when X is a threefold,

β,Htilt semistable} is empty or a union of line segments and rays.
Proof The first and third statements are in [7,Corollary3.3.3] and also in [5, Appendix B] with more details. The nested wall theorem is in [23, Theorem 3.1] and [5,Lemma 4.3]. As for the Eq. (4), by formally tensoring O(m H) on E, we may assume that H n−1 ch 1 (E) = 0. The left hand side then can be simplified as: This equals the right hand side since the zero determinant implies: The following lemma from [5] will be very useful in the technique of deforming tilt-stabilities. We list it here for the convenience of readers.

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 On may think the Brill-Noether stability condition also as the 'weak stability condition' on the heart Coh 0,H (X ) whose central charge is given by Z = −H n−2 ch 2 +i H n−1 ch 1 . 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 objectẼ is in Coh 0,H (X ) and Brill-Noether semistable.
Proof We prove the case when ν B N (E) > 0, the other case can be proved in a similar way. Note thatẼ is the canonical extension we have Hom(O X [1], Q) = 0. Therefore, we must have Hom(E, Q) = 0.
Since E is ν α,β,H -tilt stable and Q is ν α,β,H -tilt semistable with the same slope, the object E has to be a subobject of Q in Coh β,H (X ). Denote the kernel ofẼ Q by K . We then have the short exact sequence in Coh β,H (X ). By choosing sufficiently small β > 0, we have H n−1 ch 1 (Q/E), , both Q/E and K must be some direct summands of O X [1]. By the definition ofẼ, there is no non-zero map from K toẼ. Hence,Ẽ is ν ,0,H -tilt stable for sufficiently small > 0.

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 [5, Section 5], we first reduce the inequality for every tilt semistable objects to Brill-Noether stable objects.

Proposition 3.1 Let X be a smooth projective quintic threefold, and H
For any (α , β ) in W , by Lemma 2.9, the object E is ν α ,β ,H -tilt semistable. By Lemma 2.9 part (b), we have Q α ,β (E) < 0. By the assumption that , the wall W contains at least one (α 0 , β 0 ) such that β 0 is an integer.
is equivalent to say that the point (α, β) is to the right of the dashed lines Moreover, we can choose the integer β 0 such that Here the integer β 0 can be determined by the position of p H (E) as in Fig. 1. Or more precisely, the integer β 0 is in ,when ch 0 (E) > 0; ,when ch 0 (E) = 0.

Clifford type inequality for curves C 2,2,5
The generalized Clifford index theorem for curves, [9, Theorem 2.1] states that for any semistable vector bundle F over a smooth curve C with rank r and slope μ ∈ [0, g], where g is the genus of the curve, 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): when μ ∈ (0, 2); max{ 24 The bound listed above is the best result we can prove so far. As for the purpose to prove Proposition 5.2, when μ ∈ [2, 10], we only need the following weaker but neat bound.
To express those vector bundles with sharp bound more precisely, we 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 . Note that S 2,2 is a del Pezzo surface with degree 4, it can be viewed as a projective plane blown-up at 5 points. Denote 0 as the pull-back of a line in P 2 and 1 as one of the exceptional lines.
Back to the proof for the Proposition 4.1, it is enough to prove the statement for stable vector bundles. 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 . As h 0 (F) = h 0 (ι * F), 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 [14,15], we will compute h 0 (ι * F) by considering the Harder-Narasimhan factors of ι * F with respect to ν B N .

Lemma 4.4 [14, Proposition 3.4 (a)]
For each object E ∈ Coh 0,H (S 2,2 ) that is ν α,0,H -tilt stable for some α > 0, there exists δ > 0 such that there is a Harder-Narasimhan filtration for E with respect to ν α,0,H for any 0 < α < δ: The original statement in [14, Proposition 3.4 (a)] only states for K3 surfaces. But the argument only needs that there are finitely many possible classes for semistable factors, which is due to Bogomolov inequality. So it holds for every surface.
The geometric stability conditions on S 2,2 with polarization H is slightly larger than that ensured by the Bogomolov inequality. In particular, we may choose the parameter α ≤ β 2 2 .

Observation 4.6 For any torsion-free μ H -slope stable object E, we have
The stability condition is then a standard construction as that in [11] or the framework in [30, Section 2].
The following lemma explains that we can estimate the dimension of global sections for each Brill-Noether semistable factor. 322 C. Li [1] is Brill-Noether stable with slope
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.
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 O Q (Q P, respectively) unless x 1 y 1 = − n 2 ( Proof Consider the following toy model on the left in Fig. 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 ♣( 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 B N as that in Lemma 4.4: 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 x q y q is the upper bound for ν + B N (ι * F) and x p −x q x q −y q is the lower bound for ν − B N (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 O Q P as that in Fig. 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 O Q P.
We may consider when P 1 = Q or P 1 is on the line segment O Q such that

Bogomolov-Gieseker type inequality for surfaces S 2,and quintic threefolds
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 [13] will be one of the key tools to reduce Bogomolov-Gieseker type inequality for higher dimensional varieties to surfaces. (X, H ) be a polarized smooth projective variety with dimension n = 2 or 3. Let E be a coherent sheaf in Coh 0,H (X ). Suppose there exists α > 0 and m ∈ Z >0 such that
Proof Note that E(−m H) [1] is ν α,0,H -tilt stable, for any torsion sheaf T supported on a variety with codimension not less than 2, we have Hom(T, E(−m H) [1]) = 0. In particular, E is a reflexive sheaf, the singular locus of E is of codimension at least 3. For any smooth irreducible Y ∈ |m H| avoiding the singular locus, the restricted sheaf E| Y is locally free on Y . In addition, rk(E) = rk(E| Y ), H n−2 Y ch 1 (E| Y ) = m H n−1 ch 1 (E). Suppose E| Y is not semistable, then there is a destabilizing subobject F → E| Y in Coh(Y ) such that F is locally free and μ H Y (E| Y ) < μ H Y (F). Denote the embedding by ι : Y → X . Then ν α,0,H (ι * (E| Y )) = H n−2 ch 2 (ι * (E| Y )) H n−1 ch 1 (ι * (E| Y )) Therefore ι * (E| Y ) is not ν α,0 -tilt semistable. However, the object ι * (E| Y ) is the extension of E and E(−m H) [1] in Coh 0,H (X ). Since both E and E(−m H) [1] are ν α,0,H -tilt stable with the same slope in Coh 0,H (X ), any of their extension is ν α,0,H -tilt semistable. We get the contradiction, and E| Y must be μ H Y -slope semistable.
Let S 2,5 ⊂ P 4 be a smooth irreducible projective surface which is the complete intersection of a quadratic hypersurface and a quintic hypersurface. Denote H = [O S 2,5 (1)]. By the Clifford type inequality for C 2,2,5 ∈ |2H | in Proposition 4.1, we have the following stronger Bogomolov-Gieseker type inequality for stable objects in D b (S 2,5 ).