Moduli stacks and invariants of semistable objects on K3 surfaces

By Yukinobu Toda. 2007

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Abstract For a K3 surface X and its bounded derived category of coherent sheaves D ( X ) , we have the notion of stability conditions on D ( X ) in the sense of T. Bridgeland. In this paper, we show that the moduli stack of semistable objects in D ( X ) with a fixed numerical class and a phase is represented by an Artin stack of finite type over C . Then following D. Joyce's work, we introduce the invariants counting semistable objects in D ( X ) , and show that the invariants are independent of a choice of a stability condition.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e7e81c70887a6d45470810da9808d82234534317

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Citations in the corpus

Projectivity and birational geometry of Bridgeland moduli spaces (2012, unknown): “By [77, Theorem 1.4], J ( v ) does not depend on σ , and it is invariant under autoequivalences of D b ( X ).”
On the structure of instability in moduli theory (2014, background): “xample is Example 4.3 (See [HL4, Lemma 1.7.9] for the boundedness property). These hypotheses are also known to hold in several examples for Bridgeland stability conditions. Lemma 4.19 (Lemma 3.15 of [T1]). Let (Z,A) be an algebraic Bridgeland stability condition on Db(X). Assume that the functor M(”
Polynomial Bridgeland stability conditions and the large volume limit (2007, background): “f stable objects is an abstract stack (see [Lie06, Proposition 2.1.10] for a precise statement and references). However, in general it is not known whether this stack is an algebraic Artin stack; see [Tod07b] for a proof in a large class of examples. Let cbe a class in the numerical K-group. By some”
MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations (2013, background): “By [Tod08], stability is an open property in families; thus Ψ as in (d) directly induces a birational map MH(v) 99KMH′(v); in particular, (d)⇒ (a).”
Stability conditions on Kuznetsov components (2017, methodology): “In particular, the work in progress [BLM+17] uses deformation arguments to study non-emptiness of a moduli space Mσ(v) of σ-semistable objects for every class v in the numerical Grothendieck group of Ku(X) with Mukai self-pairing v2 ≥ −2; in other words, the goal is to extend the deformation argumen”
Fano varieties of cubic fourfolds containing a plane (2009, background): “3, first of all we observe that all the arguments in [32] generalize to the twisted setting.”
Lectures on Bridgeland Stability (2016, background): “23 has indeed an affirmative answer in the case of certain stability conditions on surfaces [Tod08] (or more generally, when Bridgeland stability conditions are constructed via an iterated tilting procedure [PT15]).”
Bridgeland’s stabilities on abelian surfaces (2012, background): “For a special chamber, the stability coincides with Gieseker stability ([3], [12], [2], [7]).”
Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface (2014, methodology): “In Sections 6–9, we prove the existence and projectivity of the moduli spacesMσ,Y (v): we first use results of [33, 49] to prove that the algebraic stacks Mσ,Y (v) parameterizing σ-semistable objects of Mukai vector v are non-empty finite type Artin stacks, C∗-gerbes over an algebraic space in the c”