Stability conditions in families
type: Item
instance of: scholarly article
title: Stability conditions in families
zbMATH Open document ID: 1481.14033
author: Arend Bayer
author: Martí Lahoz
author: Emanuele Macrì
author: Alexander Perry
author: Paolo Stellari
author: Howard Nuer
published in: Publications Mathématiques
publication date: 2021-09-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1902.08184
review text: The main aim of the paper is to study behaviour of Bridgeland's stability conditions in flat families of projective varieties. Let us fix a (numerical) stability condition \(\sigma\) in \(D^b (X)\) over \(S\). The authors prove that the stack parametrizing \(\sigma\)-semistable objects with fixed numerical class is algebraic of finite type over \(S\). Then one can deduce existence of moduli spaces in the relative case. Here current technology allows to get this result only in the characteristic \(0\) case. The problem with even defining the above stack is that it is not clear what a stability condition is in the relative case, i.e., for (some) derived categories of coherent sheaves on the fibers of a flat family \(X\to S\) of projective schemes over some base scheme. One of the main aims of the paper is to define and study this notion, taking into account also fibers over non-closed points of the base. The definition of stability condition in the relative case includes good openness of geometric stability in families of objects, boundedness of semistable objects etc. that are usually non-trivial theorems in the case of moduli spaces of semistable sheaves. One of the advantages of the authors' approach is that they can prove a generalization of Bridgeland's deformation theorem in the relative case. Another important result says that for a polarized flat family of smooth projective varieties one gets existence of relative stability conditions when the corresponding construction is known fiberwise. Currently this is restricted to low dimensions: \(1\), \(2\) and sometimes \(3\). They use this result to show deformation-invariance of Donaldson-Thomas invariants for some Calabi-Yau threefolds (e.g., for quintic threefolds). The authors show also some other applications of their result. One of the most important is the generalization of Mukai's theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov …
reviewed by: Adrian Langer
Mathematics Subject Classification ID: 14F08
Mathematics Subject Classification ID: 14J42
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 7395794
zbMATH Keywords: derived categories
zbMATH Keywords: Bridgeland stability conditions
zbMATH Keywords: families of varieties
zbMATH Keywords: moduli stacks
zbMATH Keywords: cubic fourfolds
MaRDI profile type: Publication
OpenAlex ID: W3160190504
arXiv ID: 1902.08184
cites work: On two rationality conjectures for cubic fourfolds
cites work: Descent for semiorthogonal decompositions
cites work: Q3844098
cites work: Cartan-Iwahori-Matsumoto decompositions for reductive groups
cites work: Good moduli spaces for Artin stacks
cites work: Vanishing and nonvanishing theorems for numerically effective bundles on complex spaces
cites work: Sheaves of t-structures and valuative criteria for stable complexes
cites work: Hodge theory and derived categories of cubic fourfolds
cites work: A short proof of the deformation property of Bridgeland stability conditions
cites work: Q3320507
cites work: Nef divisors for moduli spaces of complexes with compact support
cites work: Q3737565
cites work: Variétés kähleriennes dont la première classe de Chern est nulle
cites work: Donaldson-Thomas type invariants via microlocal geometry
cites work: The intrinsic normal cone
cites work: Mori cones of holomorphic symplectic varieties of K3 type
cites work: Arithmetically Cohen–Macaulay Bundles on Cubic Fourfolds Containing a Plane
cites work: The space of stability conditions on the local projective plane
cites work: MMP for moduli of sheaves on \(K3\)s via wall-crossing: nef and movable cones, Lagrangian fibrations
cites work: Projectivity and birational geometry of Bridgeland moduli spaces
cites work: The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds
cites work: Bridgeland Stability Conditions on Fano Threefolds
cites work: Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities
cites work: Equivalences of Triangulated Categories and Fourier-Mukai Transforms
cites work: Stability conditions on triangulated categories
cites work: Stability conditions on \(K3\) surfaces
cites work: Integral transforms and Drinfeld centers in derived algebraic geometry
cites work: On the Locus of Hodge Classes
cites work: The Hodge conjecture for fourfolds admitting a covering by rational curves
cites work: Special prime Fano fourfolds of degree 10 and index 2
cites work: Smoothness, semi-stability and alterations
cites work: Gushel-Mukai varieties: Classification and birationalities
cites work: Gushel-Mukai varieties: linear spaces and periods
cites work: On the Period Map for Polarized Hyperkähler Fourfolds
cites work: Q4549213
cites work: Hyper-Kähler fourfolds and Grassmann geometry
cites work: Q4699486
cites work: Q4342000
cites work: On a Theorem of Bogomolov on Chern Classes of Stable Bundles
cites work: Q5360829
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie). Rédigé avec la colloboration de J. Dieudonné
cites work: Q4946819
cites work: Q3579591
cites work: Derived equivalences of K3 surfaces and orientation
cites work: Tilting in abelian categories and quasitilted algebras
cites work: Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes
cites work: Birational symplectic manifolds and their deformations
cites work: Q5470192
cites work: The K3 category of a cubic fourfold
cites work: Hodge Theory of Cubic Fourfolds, Their Fano Varieties, and Associated K3 Categories
cites work: Fano manifolds of degree ten and EPW sextics
cites work: Smoothness of the moduli space of complexes of coherent sheaves on an Abelian or a projective K3 surface
cites work: A theory of generalized Donaldson-Thomas invariants

Stability conditions in families (cites work)
cites work: Singular symplectic moduli spaces

Stability conditions in families (cites work)
cites work: Q4212603

Stability conditions in families (cites work)
cites work: Symplectic structures on moduli spaces of sheaves via the Atiyah class

Stability conditions in families (cites work)
cites work: Stability conditions on product threefolds of projective spaces and Abelian varieties

Stability conditions in families (cites work)
cites work: Stability conditions on threefolds with nef tangent bundles

Stability conditions in families (cites work)
cites work: Derived categories of Gushel–Mukai varieties

Stability conditions in families (cites work)
cites work: Categorical joins

Stability conditions in families (cites work)
cites work: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants

Stability conditions in families (cites work)
cites work: Derived categories of quadric fibrations and intersections of quadrics

Stability conditions in families (cites work)
cites work: Derived categories of Fano threefolds

Stability conditions in families (cites work)
cites work: Derived Categories of Cubic Fourfolds

Stability conditions in families (cites work)
cites work: Base change for semiorthogonal decompositions

Stability conditions in families (cites work)
cites work: Calabi-Yau and fractional Calabi-Yau categories

Stability conditions in families (cites work)
cites work: Semistable sheaves in positive characteristic

Stability conditions in families (cites work)
cites work: Bogomolov's inequality for Higgs sheaves in positive characteristic

Stability conditions in families (cites work)
cites work: The moduli space of cubic fourfolds via the period map

Stability conditions in families (cites work)
cites work: On stability conditions for the quintic threefold

Stability conditions in families (cites work)
cites work: Stability conditions on Fano threefolds of Picard number 1

Stability conditions in families (cites work)
cites work: Moduli of complexes on a proper morphism

Stability conditions in families (cites work)
cites work: Remarks on the stack of coherent algebras

Stability conditions in families (cites work)
cites work: Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects

Stability conditions in families (cites work)
cites work: Twisted cubics on cubic fourfolds

Stability conditions in families (cites work)
cites work: The period map for cubic fourfolds

Stability conditions in families (cites work)
cites work: Higher Topos Theory (AM-170)

Stability conditions in families (cites work)
cites work: A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space

Stability conditions in families (cites work)
cites work: A survey of Torelli and monodromy results for holomorphic-symplectic varieties

Stability conditions in families (cites work)
cites work: Curve classes on irreducible holomorphic symplectic varieties

Stability conditions in families (cites work)
cites work: A note on the Kähler and Mori cones of hyperkähler manifolds

Stability conditions in families (cites work)
cites work: The derived category of a non-generic cubic fourfold containing a plane

Stability conditions in families (cites work)
cites work: Fourier-Mukai transforms and Bridgeland stability conditions on abelian threefolds. II.

Stability conditions in families (cites work)
cites work: Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces

Stability conditions in families (cites work)
cites work: Bridgeland stability on blow ups and counterexamples

Stability conditions in families (cites work)
cites work: Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface

Stability conditions in families (cites work)
cites work: Q3828123

Stability conditions in families (cites work)
cites work: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability

Stability conditions in families (cites work)
cites work: The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface

Stability conditions in families (cites work)
cites work: Equivalences of derived categories and \(K3\) surfaces

Stability conditions in families (cites work)
cites work: Triangulated categories of singularities and equivalences between Landau-Ginzburg models

Stability conditions in families (cites work)
cites work: Lagrangian embeddings of cubic fourfolds containing a plane

Stability conditions in families (cites work)
cites work: Noncommutative homological projective duality

Stability conditions in families (cites work)
cites work: Q5421225

Stability conditions in families (cites work)
cites work: Moduli of Bridgeland semistable objects on 3-folds and Donaldson-Thomas invariants

Stability conditions in families (cites work)
cites work: Q3934506

Stability conditions in families (cites work)
cites work: A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold

Stability conditions in families (cites work)
cites work: Counterexample to the Generalized Bogomolov–Gieseker Inequality for Threefolds

Stability conditions in families (cites work)
cites work: Moduli of representations of the fundamental group of a smooth projective variety. I

Stability conditions in families (cites work)
cites work: Stability conditions under change of base field

Stability conditions in families (cites work)
cites work: A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on \(K3\) fibrations.

Stability conditions in families (cites work)
cites work: Moduli stacks and invariants of semistable objects on \(K3\) surfaces

Stability conditions in families (cites work)
cites work: Some aspects of the Hodge conjecture

Stability conditions in families (cites work)
cites work: Q2950764

Stability conditions in families (cites work)
cites work: Hyper-Kähler compactification of the intermediate Jacobian fibration of a cubic fourfold: The twisted case

Stability conditions in families (cites work)
cites work: Some notes on the moduli of stable sheaves on elliptic surfaces

Stability conditions in families (cites work)
cites work: Moduli spaces of stable sheaves on abelian surfaces.

Stability conditions in families (cites work)
cites work: Q4116549

Stability conditions in families (DOI)
DOI: 10.1007/S10240-021-00124-6