Bridgeland-stable moduli spaces for $K$-trivial surfaces

Abstract

ABSTRACT. We give a natural family of Bridgeland stability conditions on the derived category of a smooth projective complex surface S and describe "wallcrossing behavior" for objects with the same invariants as O C (H) when H generates Pic(S) and C ∈ |H|. If, in addition, S is a K3 or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgelandstable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural generalization of Thaddeus' stable pairs for curves embedded in the moduli spaces.

Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture

“…[Tod09a, Lemma 2.7]). Further, the construction of stability conditions for surfaces (see [Bri08,ABL07]) needs the Bogomolov-Gieseker inequality for slope-stable bundles and the Hodge Index theorem. The methods of [BM11] imply an even closer relationship: knowing the set of possible numerical central charges Z for which skyscraper sheaves of points k(x) are stable is essentially equivalent to knowing the set of Chern characters of slope-semistable bundles for any polarization of X.…”

Lectures on Bridgeland Stability

“…The main result is the following (see [Bri08,AB13,BM11]). We will choose Λ = K num (X) and v as the Chern character map as in Example 5.7.…”

“…As another application, we will give a proof of Kodaira vanishing for surfaces using tilt stability. The argument was first pointed out in [AB13]. While it is a well-known argument by Mumford that Kodaira vanishing in the surface case is a consequence of Bogomolov's inequality, this proof follows a slightly different approach.…”

“…The idea of using this operation to construct Bridgeland stability conditions is due to Bridgeland in [Bri08], in the case of K3 surfaces. The extension to general smooth projective surfaces is in [AB13].…”

“…This construction originated in [Bri08] in the case of K3 surfaces. Arcara and Bertram realized that the construction can be generalized to any surface by using the Bogomolov Inequality in [AB13]. The proof of the support property is in [BM11,BMT14,BMS14].…”