Moduli of complexes on a proper morphism

Abstract

Abstract. Given a proper morphism X → S, we show that a large class of objects in the derived category of X naturally form an Artin stack locally of finite presentation over S. This class includes S-flat coherent sheaves and, more generally, contains the collection of all S-flat objects which can appear in the heart of a reasonable sheaf of t-structures on X. In this sense, this is the Mother of all Moduli Spaces (of sheaves). The proof proceeds by studying the finite presentation properties, deformation theory, and Grothendieck existence theorem for objects in the derived category, and then applying Artin's representability theorem.

Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

“…The deformation theory of objects of the derived category of coherent sheaves on a smooth projective family has been studied in [13,14]. These papers produce a class which is the obstruction to deforming a complex sideways over an infinitesimal deformation.…”

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

“…To show that U is open, it suffices to show that U is constructible and stable under generization. By standard results (e.g., 2.1.3 and 2.1.4 of [7]), the locus in B over which the cohomology of the geometric fibers of E is concentrated in degrees −1 and 0 is open. Thus, we may assume from the start that H i (E) = 0 unless i ∈ {−1, 0}.…”

“…We recall the main theorem of [7]. This is the Mother of All Moduli Spaces: it contains the hearts of all of the sheaves of t-structures on X.…”

“…More precisely, we will show that there is an open subscheme U such that for a point b → B, the derived base change E b is in D (T,F ) if and only if b factors through U . It follows from 2.2.1 of [7] and 8.10.5 of [5] that we may assume B is Noetherian.…”

Lectures on Bridgeland Stability

“…We follow [Ina02,Lie06]. Let B be a scheme locally of finite type over C. We denote the unbounded derived category of quasi-coherent sheaves by D(Qcoh(X × B)).…”