Stability conditions in families

Abstract

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

Fullness of exceptional collections via stability conditions -- A case study: the quadric threefold

“…The spinor bundle S is slope stable (see, e.g, [31]), so it is is σ α,β -stable for α sufficiently big and any β by [4,Proposition 2.13]. Since there are no walls in the (α, β)-semiplane (Lemma 5.7), we see that S is also σ α,β -stable for any α, β.…”

“…Fano threefolds of Picard rank 1 are known to have stability conditions ( [26], preceeded by [28] for the particular case of P 3 and by [32] for the case of the quadric threefold Q 3 considered in this article). We preferred not to use this stronger result and only use the weak stability condition (2.1) on D b (Q 3 ) to induce a stability condition on Ku(Q 3 ) following [4], as only a stability condition on the residual category is relevant to the constructions in this article, and weak stability conditions on the ambient category are much easier to have with respect to actual stability conditions. For instance, (2.1) defines a weak stability condition on every smooth projective variety, regardless of its dimension.…”

“…In the same flavor, Bridgeland stability conditions on the derived category of an algebraic variety X are often used to investigate the geometric properties of moduli spaces of sheaves and complexes of sheaves on X, and their very existence is an intense object of study on threefolds and higher dimensional varieties [11,12,13,14]. Recently, Bayer-Lahoz-Macrì-Stellari [4] have showed how to induce a stability condition on the Kuznetsov component of a projective variety from a weak stability condition on its hosting derived category, provided that certain conditions are satisfied. These conditions are notably satisfied by Fano threefolds of Picard rank 1.…”

“…A stability condition, including a "positivity condition" for nonzero objects, notably forbids the presence of phantom subcategories (or at least, as we will show, of phantomic summands, which is enough for our purposes), thus remarkably simplifying proofs of fullness. In particular we will focus our attention on a simple example, while at the same time indicating how a few others can at least in principle be described in a similar fashion: the index 3 Fano threefold, i.e., a smooth quadric threefold Q 3 in P 4 . Quadrics are actually one of those few notable cases (among, e.g., projective spaces and Grassmannians) where exhibiting a full exceptional collection does not require somehow sophisticated techniques as stability conditions, see [18,19].…”

On the Bogomolov-Gieseker inequality in positive characteristic

“…One big part of the proof is the argument similar to the one in [Lan04], where he proves the BG inequality in positive chararcteristic, with the modified term depending on ch 0 (E) 4 . In addition to his arguments, the key fact we use is the"invariance" of the BG inequality under blowups and change of polarizations, discussed in [BLMS17,Lan16]. According to these invariance, we are able to proceed by induction on ch 0 (E) in the birational equivalence class of a surface, rather than the whole category of surfaces.…”