The wall-crossing behavior for Bridgeland’s stability conditions on abelian and K3 surfaces

2018 ◽  
Vol 2018 (735) ◽  
pp. 1-107 ◽  
Author(s):  
Hiroki Minamide ◽  
Shintarou Yanagida ◽  
Kōta Yoshioka
Keyword(s):  
Stability Condition ◽  
Derived Category ◽  
The Other ◽  
Abelian Surface ◽  
Stability Conditions ◽  
Abelian Surfaces ◽  
Coherent Sheaves ◽  
Fundamental Properties ◽  
Wall Crossing ◽  
Wall Crossing Formula

AbstractThe wall-crossing behavior for Bridgeland’s stability conditions on the derived category of coherent sheaves on K3 or abelian surface is studied. We introduce two types of walls. One is called the wall for categories, where thet-structure encoded by stability condition is changed. The other is the wall for stabilities, where stable objects with prescribed Mukai vector may get destabilized. Some fundamental properties of walls and chambers are studied, including the behavior under Fourier–Mukai transforms. A wall-crossing formula of the counting of stable objects will also be derived. As an application, we will explain previous results on the birational maps induced by Fourier–Mukai transforms on abelian surfaces. These transformations turns out to coincide with crossing walls of certain property.

scholarly journals Stability conditions and birational geometry of projective surfaces

2014 ◽  
Vol 150 (10) ◽  
pp. 1755-1788 ◽  
Author(s):  
Yukinobu Toda
Keyword(s):  
Moduli Spaces ◽  
Minimal Model ◽  
Derived Category ◽  
Stability Conditions ◽  
Birational Geometry ◽  
Projective Surface ◽  
Model Program ◽  
Minimal Model Program ◽  
Coherent Sheaves ◽  
Projective Surfaces

AbstractWe show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.

scholarly journals Stability conditions for generic K3 categories

2008 ◽  
Vol 144 (1) ◽  
pp. 134-162 ◽  
Author(s):  
Daniel Huybrechts ◽  
Emanuele Macrì ◽  
Paolo Stellari
Keyword(s):  
K3 Surfaces ◽  
Derived Categories ◽  
Picard Group ◽  
Stability Conditions ◽  
Abelian Surfaces ◽  
Coherent Sheaves ◽  
The Stability ◽  
Stability Manifold

AbstractA K3 category is by definition a Calabi–Yau category of dimension two. Geometrically K3 categories occur as bounded derived categories of (twisted) coherent sheaves on K3 or abelian surfaces. A K3 category is generic if there are no spherical objects (or just one up to shift). We study stability conditions on K3 categories as introduced by Bridgeland and prove his conjecture about the topology of the stability manifold and the autoequivalences group for generic twisted projective K3, abelian surfaces, and K3 surfaces with trivial Picard group.

scholarly journals Formality conjecture for K3 surfaces

2019 ◽  
Vol 155 (5) ◽  
pp. 902-911 ◽  
Author(s):  
Nero Budur ◽  
Ziyu Zhang
Keyword(s):  
Line Bundle ◽  
Stability Condition ◽  
Ample Line Bundle ◽  
Main Tool ◽  
K3 Surfaces ◽  
Derived Category ◽  
K3 Surface ◽  
Coherent Sheaves ◽  
Dg Algebra ◽  
Complex Projective

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra$\operatorname{RHom}^{\bullet }(F,F)$is formal for any sheaf$F$polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

scholarly journals On K3 Surface Quotients of K3 or Abelian Surfaces

2017 ◽  
Vol 69 (02) ◽  
pp. 338-372
Author(s):  
Alice Garbagnati
Keyword(s):  
Abelian Group ◽  
Moduli Space ◽  
Minimal Model ◽  
K3 Surfaces ◽  
Rational Curves ◽  
The Other ◽  
Abelian Surface ◽  
K3 Surface ◽  
Abelian Surfaces ◽  
Special Cases

Abstract The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group G (respectively of a K3 surface by an Abelian group G) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the K3 surfaces that are (rationally) G-covered by Abelian or K3 surfaces (in the latter case G is an Abelian group). When G has order 2 or G is cyclic and acts on an Abelian surface, this result is already known; we extend it to the other cases. Moreover, we prove that a K3 surface XG is the minimal model of the quotient of an Abelian surface by a group G if and only if a certain configuration of rational curves is present on XG . Again, this result was known only in some special cases, in particular, if G has order 2 or 3.

scholarly journals On a family of (1,7)-polarised abelian surfaces

2004 ◽  
Vol 95 (2) ◽  
pp. 181 ◽  
Author(s):  
Alfio Marini
Keyword(s):  
Moduli Space ◽  
Level Structure ◽  
The Other ◽  
Abelian Surface ◽  
Abelian Surfaces

We study the geometry of the moduli space of $(1,7)$-polarised abelian surfaces with canonical level structure in detail. In particular we describe the locus where the syzygies of the embedded Heisenberg-invariant abelian surface degenerate, and relate this to the other known descriptions of the moduli space in question.

scholarly journals On the issue of the modern system of tax law

2020 ◽  
Author(s):  
G.F. Cel'niker ◽  
N.A. Fityunina ◽  
S.A. Zvyaginceva
Keyword(s):  
Russian Federation ◽  
Derived Category ◽  
The Other ◽  
Tax Law ◽  
Law System ◽  
Separate Branch ◽  
Modern System ◽  
The Russian Federation ◽  
The One ◽  
Functional Purpose

The article reveals the features of the tax law system, which is considered, on the one hand, as an Autonomous, separate branch of law in the system of branches of law of the Russian Federation, and on the other, as a derived category from the norms that determine financial law and, thus, are a sub-branch by their functional purpose. The criteria on the basis of which it seems appropriate to allocate institutions in the tax law system are highlighted. The General and special parts of tax law are characterized through the prism of their Conditioned norms.

scholarly journals A note on thick subcategories of stable derived categories

2013 ◽  
Vol 212 ◽  
pp. 87-96
Author(s):  
Henning Krause ◽  
Greg Stevenson
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Complete Intersections ◽  
Line Bundles ◽  
Finitely Generated ◽  
Exact Category ◽  
Coherent Sheaves ◽  
Noetherian Scheme ◽  
Finitely Generated Modules ◽  
Local Complete Intersections

AbstractFor an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection, we classify thick subcategories of finitely generated modules over strict local complete intersections and produce generators for the category of coherent sheaves on a separated Noetherian scheme with an ample family of line bundles.

scholarly journals Completing perfect complexes

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1387-1427 ◽  
Author(s):  
Henning Krause
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Direct Construction ◽  
Coherent Sheaves ◽  
Perfect Complexes ◽  
Coherent Ring ◽  
Abelian Categories ◽  
Finitely Presented

Abstract This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

scholarly journals A wall-crossing formula for degrees of Real central projections

2014 ◽  
Vol 25 (04) ◽  
pp. 1450038 ◽  
Author(s):  
Christian Okonek ◽  
Andrei Teleman
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Pole Placement ◽  
Real Projective Space ◽  
Central Projections ◽  
Wall Crossing ◽  
Subspace Problem ◽  
Wall Crossing Formula ◽  
Real Subspace ◽  
Crucial Assumption

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

Fourier-Mukai Transforms in Algebraic Geometry

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Algebraic Geometry ◽  
Smooth Projective Variety ◽  
Derived Category ◽  
Point Of View ◽  
Basic Knowledge ◽  
Research Directions ◽  
Coherent Sheaves ◽  
Geometric Point ◽  
Singular Cohomology

This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

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