Moduli of complexes on a proper morphism

By Max Lieblich. 2005

222 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1a36db12e8d947bebe53ac166fb42d2e3b59287e

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Citations in the corpus

Polynomial Bridgeland stability conditions and the large volume limit (2007, background): “is known that the moduli problem of stable objects is an abstract stack (see [ Lie06 , Proposition 2.1.10] for a precise statement and references).”
Hodge theory and derived categories of cubic fourfolds (2012, unknown): “amely, replacing M t by a Zariski open neighbourhood of X t ˆM t gives a Zariski open in the Artin stack of all objects (with no negative self-Exts) of D(X t). This stack does behave well in families [30], giving a limiting stack of objects in D(X 1). These include P x; x2X 1, so this stack contains”
Higher and derived stacks: a global overview (2009, methodology): “It is easy to see that, as a complex, Lx is naturally quasi-isomorphic to TxF[−1], the shifted tangent complex of F at x. Therefore, we obtain a natural structure of dg-Lie algebra (or at least L∞- Lie algebra) on TxF[−1].”
Moduli stacks and invariants of semistable objects on K3 surfaces (2007, background): “In Section 3, the work of Lieblich [23] would help us.”
Derived equivalences of K3 surfaces and orientation (2007, methodology): “The proof of the theorem, which makes use of Lieblich’s obstruction in [25], can be found in [19, App.”
Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces (2018, background): “ a scheme Bwhich is locally of ﬁnite type over Kto the groupoid M(B) := ( E∈ DB-perf : Exti(E| X×{b},E|X×{b}) = 0, for all i<0 and all geometric points b∈ B ) . The following is the main result in [Lie06]: Theorem 5.10 (Lieblich). The functor M is an Artin stack, locally of ﬁnite type, locally qu”
Existence of moduli spaces for algebraic stacks (2018, background): “C), MA is an open sub-functor of the moduli functor Db pug(X) of universally glueable relatively perfect complexes on X and is hence an algebraic stack locally of ﬁnite type with aﬃne diagonal over k [Lie06], [Stacks, Tag 0DPW]. By [Pol07, Prop. 3.3.7], MA as deﬁned above agrees with the moduli func”
Moduli of objects in dg-categories (2005, methodology): “Contrary to [19] our proof does not use Artin’s representability criterion, and also provides descriptions of sub-stacks of finite type (see Corollary 3.”
The K3 category of a cubic fourfold (2015, background): “Note that at the very end of the argument one needs to apply a result of Lieblich [Lie06] saying that the space of objects with no negative self-Exts in the derived category is an Artin stack of locally finite presentation.”
Fourier-Mukai partners of K3 surfaces in positive characteristic (2011, methodology): “K) parametrizing the formal lift Xis Φ−1(Fil2 H2 Y /K)). Since Φ−1(Fil2 H2 Y /K)) = Fil 2 X H 2(X/K), we conclude that X= Xb W. Applying the Grothendieck Existence Theorem for perfect complexes as in [L], we get the desired lift PW ∈ D(XW ×W YW ). Remark 8.2. If we had an integral version of the Muk”
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”
Bridgeland-stable moduli spaces for $K$-trivial surfaces (2007, background): “It is a standard result (Corollaire 2.1.23 of [2] or Theorem 2.1.9 of [1]) that Dpug(X/S) is a stack on the fpqc topology on the category of S-schemes. We recall the main theorem of [ 7 ].”