On stability conditions for the quintic threefold

Invent. math. https://doi.org/10.1007/s00222-019-00888-z On stability conditions for the quintic threefold Chunyi Li1 Received: 29 October 2018 / Accepted: 11 April 2019 © The Author(s) 2019 Abstract We study the Clifford type inequality for a particular type of curves C2,2,5 , 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, Macrì, Stellari and Toda, we construct an open subset of stability conditions on every smooth 4. quintic threefold in PC Mathematics Subject Classification Primary 14F05; Secondary 14J32 · 18E30 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Organisation and Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Background: tilt-stability condition and wall-crossing . . . . . . . . . . . . . . . . . . . 2.1 Stability condition: notations and conventions . . . . . . . . . . . . . . . . . . . . . 2.2 Recollection of lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Proof for the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Clifford type inequality for curves C2,2,5 . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Bogomolov–Gieseker type inequality for surfaces S2,5 and quintic threefolds . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Chunyi Li https://sites.google.com/site/chunyili0401/ 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK 123 C. Li 1 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: Theorem 1.1 (Theorem 5.5) Let F be a torsion free μ H -slope semistable 2 1 (F) sheaf on a smooth quintic threefold (X, H ). Suppose HH 3ch ∈ [−1, 1], rk(F) then  ⎧  2    H ch1 (F)  1 2 , ⎪ ch (F) when H −  ∈ [0, 41 ];  ⎪ 1 ⎪ H 3 rk(F)   ⎨ 2   2 1 (F)  H ch2 (F) ≤ 21  H 2 ch1 (F) − 45 rk(E), when  HH 3ch  ∈ [ 41 , 43 ]; rk(F)   ⎪   ⎪ ⎪ ⎩ 3  H 2 ch1 (F) − 5rk(E), when  H 23ch1 (F)  ∈ [ 3 , 1]. 2 4 H rk(F)    2  2  H ch1 (F)  1  H ch1 (F)  The ‘=’ can hold only when  H 3 rk(F)  ∈ 4 Z. Moreover, when  H 3 rk(F)  ∈ 1 9 [0, 10 ] ∪ [ 10 , 1], we have the following stronger bound: H ch2 (F) ≤   2 2 3 (H ch1 (F)) −  H 2 ch1 (F). 3 2 H rk(F) 123 On stability conditions for the quintic threefold 1 (F) In a special case that when HH 3ch = − 21 , we have (F)H ≥ 1.25rk(F)2 , rk(F) 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. 2 Theorem 1.2 (Theorem 2.8) Conjecture 4.1 in [5] holds for smooth quintic threefolds when the parameters satisfy α 2 + (β − β − 21 )2 > 41 . 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. (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 νσ (E) := − Re Z ([E]) . ImZ ([E]) (b) With respect to the slope-stability νσ , each non-zero object E ∈ A admits a unique Harder–Narasimhan filtration: 0 = E0 ⊂ E1 ⊂ · · · ⊂ Em = E such that: each quotient Fi := E i /E i−1 is μσ -slope semistable with νσ (F1 ) > νσ (F2 ) > · · · > νσ (Fm ). We set νσ+ (E) := νσ (F1 ) and νσ− (E) := νσ (Fm ). (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 123 C. Li (minimum) slope of its Harder–Narasimhan factors. The first tilting-heart Cohβ,H (X ) ⊂ D b (X ) with parameter β ∈ R is the extension-closure Tβ,H , F H,β [1] , where + Tβ,H = {E ∈ Coh(X )|μ− H (E) > β}; Fβ,H = {E ∈ Coh(X )|μ H (E) ≤ β}. Given α ∈ R>0 , we may define the tilt-slope function for objects in Cohβ,H (X ) as follows: for an object E ∈ Cohβ,H (X ), its tilt-slope function is defined as  (E) := να,β,H ⎧ 2 ⎨ H chβ2 H (E)− α2 H 3 ch0 (E) , ⎩ +∞, βH H 2 ch1 (E) βH when H 2 ch1 (E) > 0; βH when H 2 ch1 (E) (...truncated)