scholarly journals Height of exceptional collections and Hochschild cohomology of quasiphantom categories

2015 ◽  
Vol 2015 (708) ◽  
Author(s):  
Alexander Kuznetsov
Keyword(s):  
Projective Variety ◽  
Hochschild Cohomology ◽  
Smooth Projective Variety ◽  
Derived Category ◽  
Coherent Sheaves

AbstractWe define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety

POSTNIKOV-STABILITY FOR COMPLEXES ON CURVES AND SURFACES

2012 ◽  
Vol 23 (02) ◽  
pp. 1250048 ◽  
Author(s):  
GEORG HEIN ◽  
DAVID PLOOG
Keyword(s):  
Moduli Space ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Derived Category ◽  
Stable Bundles ◽  
Coherent Sheaves ◽  
Curves And Surfaces

We present a novel notion of stable objects in the derived category of coherent sheaves on a smooth projective variety. As one application we compactify a moduli space of stable bundles using genuine complexes.

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.

scholarly journals Perverse Coherent Sheaves on Blow-ups at Codimension 2 Loci

2019 ◽  
Author(s):  
Naoki Koseki
Keyword(s):  
Moduli Space ◽  
Hilbert Scheme ◽  
Projective Variety ◽  
Blow Up ◽  
Smooth Projective Variety ◽  
Coherent Sheaves ◽  
Codimension Two ◽  
Higher Dimensional ◽  
Stable Sheaves ◽  
Closed Subvariety

Abstract Let $f \colon X \to Y$ be the blow-up of a smooth projective variety $Y$ along its codimension two smooth closed subvariety. In this paper, we show that the moduli space of stable sheaves on $X$ and $Y$ are connected by a sequence of flip-like diagrams. The result is a higher dimensional generalization of the result of Nakajima and Yoshioka, which is the case of $\dim Y=2$. As an application of our general result, we study the birational geometry of the Hilbert scheme of two points.

scholarly journals Derived categories of Gushel–Mukai varieties

2018 ◽  
Vol 154 (7) ◽  
pp. 1362-1406 ◽  
Author(s):  
Alexander Kuznetsov ◽  
Alexander Perry
Keyword(s):  
Hochschild Cohomology ◽  
Grothendieck Group ◽  
Derived Categories ◽  
Hochschild Homology ◽  
Derived Category ◽  
K3 Surface ◽  
Coherent Sheaves ◽  
Special Case ◽  
Rationality Problem ◽  
Cubic Fourfold

We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel–Mukai fourfold in more detail. Namely, we show this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational Gushel–Mukai fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel–Mukai varieties, which was one of the main motivations for this work.

scholarly journals Geometric collections and Castelnuovo–Mumford regularity

2007 ◽  
Vol 143 (3) ◽  
pp. 557-578 ◽  
Author(s):  
L. COSTA ◽  
R. M. MIRÓ–ROIG
Keyword(s):  
Projective Variety ◽  
Projective Spaces ◽  
Smooth Projective Variety ◽  
Coherent Sheaf ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Quadric Hypersurface ◽  
Basic Facts ◽  
Formal Properties ◽  
Smooth Projective Varieties

AbstractThe paper begins by overviewing the basic facts on geometric exceptional collections. Then we derive, for any coherent sheaf $\cF$ on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to $\cF$ and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo–Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties X with a geometric collection σ. We define the notion of regularity of a coherent sheaf $\cF$ on X with respect to σ. We show that the basic formal properties of the Castelnuovo–Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on $\PP^n$ and for a suitable geometric collection of coherent sheaves on $\PP^n$ both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface $Q_n \subset \PP^{n+1}$ (n odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo–Mumford regularity of their extension by zero in $\PP^{n+1}$.

scholarly journals Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

2021 ◽  
Author(s):  
Lie Fu ◽  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Projective Variety ◽  
Arbitrary Dimension ◽  
Smooth Projective Variety ◽  
The Other ◽  
Fano Variety ◽  
Fano Varieties ◽  
Intersection Product ◽  
Canonical Class ◽  
Main Input

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

scholarly journals Vector bundles trivialized by proper morphisms and the fundamental group scheme

2010 ◽  
Vol 10 (2) ◽  
pp. 225-234 ◽  
Author(s):  
Indranil Biswas ◽  
João Pedro P. Dos Santos
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Surjective Morphism ◽  
Finite Vector

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

scholarly journals A Spectral Sequence for the Hochschild Cohomology of a Coconnective DGA

2013 ◽  
Vol 112 (2) ◽  
pp. 182 ◽  
Author(s):  
Shoham Shamir
Keyword(s):  
Spectral Sequence ◽  
Loop Space ◽  
Hochschild Cohomology ◽  
Derived Category ◽  
Orientable Manifold ◽  
Support Varieties ◽  
Mod P

A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence presented in [7] for the computation of the loop homology of a closed orientable manifold. Using this spectral sequence we identify a class of spaces for which the Hochschild cohomology of their mod-$p$ cochain algebra is Noetherian. This implies, among other things, that for such a space the derived category of mod-$p$ chains on its loop-space carries a theory of support varieties.

scholarly journals A Krull-Schmdit type theorem for coherent sheaves

2014 ◽  
Vol 22 (2) ◽  
pp. 51-56
Author(s):  
A. S. Argáez
Keyword(s):  
Finite Group ◽  
Projective Variety ◽  
Type Theorem ◽  
Coherent Sheaf ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Coherent Sheaves ◽  
A Finite Group ◽  
Natural Decomposition

AbstractLet X be projective variety over an algebraically closed field k and G be a finite group with g.c.d.(char(k), |G|) = 1. We prove that any representations of G on a coherent sheaf, ρ : G → End(ℰ), has a natural decomposition ℰ ≃ ⊕ V ⊗k ℱV, where G acts trivially on ℱV and the sum run over all irreducible representations of G over k.

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.

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