Projectivity and Birational Geometry of Bridgeland moduli spaces
We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wall-crossing for Bridgeland-stability and the minimal model program for the moduli space.
We give three applications of our method for classical moduli spaces of sheaves on a K3 surface:
1. We obtain a region in the ample cone in the moduli space of Gieseker-stable sheaves only depending on the lattice of the K3.
2. We determine the nef cone of the Hilbert scheme of points on a K3 surface of Picard rank one when is large compared to the genus.
3. We verify the “Hassett-Tschinkel/Huybrechts/Sawon” conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.
Key words and phrases:Bridgeland stability conditions, Derived category, Moduli spaces of complexes, Mumford-Thaddeus principle
2010 Mathematics Subject Classification:14D20, (Primary); 18E30, 14J28, 14E30 (Secondary)
In this paper, we give a canonical construction of determinant line bundles on any moduli space of Bridgeland-semistable objects. Our construction has two advantages over the classical construction for semistable sheaves: our divisor class varies naturally with the stability condition, and we can show that our divisor is automatically nef.
This also explains a picture envisioned by Bridgeland, and observed in examples by Arcara-Bertram and others, that relates wall-crossing under a change of stability condition to the birational geometry and the minimal model program of . As a result, we can deduce properties of the birational geometry of from wall-crossing; this leads to new results even when coincides with a classical moduli space of Gieseker-stable sheaves.
Moduli spaces of complexes
Moduli spaces of complexes first appeared in [Bridgeland:Flop]: the flop of a smooth threefold can be constructed as a moduli space parameterizing perverse ideal sheaves in the derived category of . Recently, they have turned out to be extremely useful in Donaldson-Thomas theory; see [Toda:Survey] for a survey.
Ideally, the necessary notion of stability of complexes can be given in terms of Bridgeland’s notion of a stability condition on the derived category, introduced in [Bridgeland:Stab]. Unlike other constructions (as in [Toda:limit-stable, large-volume]), the space of Bridgeland stability conditions admits a well-behaved wall and chamber structure: the moduli space of stable objects with given invariants remains unchanged unless the stability condition crosses a wall. However, unlike Gieseker-stability for sheaves, Bridgeland stability is not a priori connected to a GIT problem. As a consequence, while established methods ([Inaba, Lieblich:mother-of-all, Toda:K3, Abramovich-Polishchuk:t-structures]) can prove existence of moduli spaces as algebraic spaces or Artin stacks, there are so far only ad-hoc methods to prove that they are projective, or to construct coarse moduli spaces.
In this paper, we propose a solution to this problem by constructing a family of numerically positive divisor classes on any moduli space of Bridgeland-stable complexes.
A family of nef divisors on Bridgeland-moduli spaces
Let be a smooth projective variety over . We denote by its bounded derived category of coherent sheaves, and by the space of Bridgeland stability conditions on , see Section LABEL:sec:Bridgeland. We refer to p. LABEL:subsec:notation for an overview of notations.
Let be a stability condition, and a choice of numerical invariants. Assume that we are given a family of -semistable objects of class parameterized by a proper algebraic space of finite type over ; for example, could be a fine moduli space of stable objects. We define a numerical Cartier divisor class as follows: for any projective integral curve , we set
where is the Fourier-Mukai functor with kernel , and is the structure sheaf of . It is easy to prove that (1) defines a numerical divisor class . Our main result, the Positivity Lemma LABEL:Positivity, implies the positivity of this divisor:
The divisor class is nef. Additionally, we have if and only if for two general points , the corresponding objects are -equivalent.
(Two semistable objects are -equivalent if their Jordan-Hölder filtrations into stable factors of the same phase have identical stable factors.) The class can also be given as a determinant line bundle. The main advantage of our construction is that we can show its positivity property directly, without using GIT; instead, the proof is based on a categorical construction by Abramovich and Polishchuk [Abramovich-Polishchuk:t-structures, Polishchuk:families-of-t-structures]. Our construction also avoids any additional choices: it depends only on .
Chambers in and the nef cone of the moduli spaces
Consider a chamber for the wall-and-chamber decomposition with respect to ; then (assuming its existence) the moduli space of -stable objects of class is constant for . Also assume for simplicity that it admits a universal family . Theorem 1.1 yields an essentially linear map
This immediately begs for the following two questions:
- Question 1:
Do we actually have ?
- Question 2:
What will happen at the walls of ?
K3 surfaces: Overview
While our above approach is very general, we now restrict to the case where is a smooth projective K3 surface. In this situation, Bridgeland described (a connected component of) the space of stability conditions in [Bridgeland:K3], and Toda proved existence results for moduli spaces in [Toda:K3]; see Section LABEL:sec:reviewK3. The following paraphrases a conjecture proposed by Bridgeland in Section 16 of the arXiv-version of [Bridgeland:K3]:
Conjecture 1.2 (Bridgeland).
Given a stability condition on a K3 surface, and a numerical class , there exists a coarse moduli space of -semistable complexes with class . Changing the stability condition produces birational maps between the coarse moduli spaces.
Our main results give a partial proof of this conjecture, and answers to the above questions: Theorem 1.3 answers Question 1 and proves existence of coarse moduli spaces; Theorem 1.4 partially answers Question 2 and partially proves the second statement of the conjecture. They also give a close relation between walls in , and walls in the movable cone of the moduli space separating nef cones of different birational models.
Projectivity of the moduli spaces
Assume that is generic, which means that it does not lie on a wall with respect to .
Let be a smooth projective K3 surface, and let . Assume that the stability condition is generic with respect to . Then:
The coarse moduli space of -semistable objects with Mukai vector exists as a normal projective irreducible variety with -factorial singularities.
The assignment (1) induces an ample divisor class on .
This generalizes [MYY2, Theorem 0.0.2], which shows projectivity of in the case where has Picard rank one.
Wall-crossing and birational geometry of the moduli spaces
We also use Theorem 1.1 to study the wall-crossing behavior of the moduli space under deformations of .
Assume that is a primitive class. Let be a wall of the chamber decomposition for . Let be a generic point of , and let be two stability conditions nearby on each side of the wall. By Theorem 1.3 and its proof, they are smooth projective Hyperkähler varieties. Since being semistable is a closed condition in , the (quasi-)universal families on are also families of -semistable objects. Theorem 1.1 also applies in this situation, and thus produces nef divisor classes on . In Section LABEL:sec:flops, we prove:
Let be a smooth projective K3 surface, and be a primitive class.
The classes are big and nef, and induce birational contraction morphisms
where are normal irreducible projective varieties.
If there exist -stable objects, and if their complement in has codimension at least two , then one of the following two possibilities holds:
Both and are ample, and the birational map
obtained by crossing the wall in , extends to an isomorphism.
Neither nor is ample, and is the flop induced by : we have a commutative diagram of birational maps