Cohomology of quasi-coherent sheaves over projective schemes

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

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EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s147474802100027x ◽

2021 ◽

pp. 1-66

Author(s):

Tom Bachmann ◽

Kirsten Wickelgren

Keyword(s):

Commutative Algebra ◽

Vector Bundles ◽

Euler Class ◽

Complete Intersections ◽

Bilinear Forms ◽

Euler Numbers ◽

Coherent Sheaves ◽

Linear Subspaces ◽

Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

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A spectral incarnation of affine character sheaves

Compositio Mathematica ◽

10.1112/s0010437x17007278 ◽

2017 ◽

Vol 153 (9) ◽

pp. 1908-1944

Author(s):

David Ben-Zvi ◽

David Nadler ◽

Anatoly Preygel

Keyword(s):

Integral Transforms ◽

Companion Paper ◽

Langlands Program ◽

Coherent Sheaves ◽

Singular Support ◽

Geometric Langlands Program ◽

Descent Theory ◽

Geometric Langlands ◽

Support Conditions ◽

Character Sheaves

We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions, which appear in recent advances in the geometric Langlands program. The key technical tools in our arguments are a new descent theory for coherent sheaves or ${\mathcal{D}}$-modules with prescribed singular support and the theory of integral transforms for coherent sheaves developed in the companion paper by Ben-Zvi et al. [Integral transforms for coherent sheaves, J. Eur. Math. Soc. (JEMS), to appear].

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Infinite root stacks and quasi-coherent sheaves on logarithmic schemes

Proceedings of the London Mathematical Society ◽

10.1112/plms.12109 ◽

2018 ◽

Vol 116 (5) ◽

pp. 1187-1243 ◽

Cited By ~ 6

Author(s):

Mattia Talpo ◽

Angelo Vistoli

Keyword(s):

Coherent Sheaves

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Vector bundles associated to Lie algebras

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2014-0007 ◽

2016 ◽

Vol 2016 (716) ◽

Cited By ~ 1

Author(s):

Jon F. Carlson ◽

Eric M. Friedlander ◽

Julia Pevtsova

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Vector Bundles ◽

Coherent Sheaves ◽

Restricted Lie Algebra ◽

Finite Dimensional

AbstractWe introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra

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Coherent Sheaves

Universitext - Algebraic Surfaces and Holomorphic Vector Bundles ◽

10.1007/978-1-4612-1688-9_3 ◽

1998 ◽

pp. 25-58

Author(s):

Robert Friedman

Keyword(s):

Coherent Sheaves

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A Krull-Schmdit type theorem for coherent sheaves

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2014-0030 ◽

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.

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