# Variation of geometric invariant theory quotients and derived categories

@article{Ballard2019VariationOG, title={Variation of geometric invariant theory quotients and derived categories}, author={Matthew Robert Ballard and David Favero and Ludmil Katzarkov}, journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)}, year={2019} }

We study the relationship between derived categories of factorizations on gauged Landau–Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are comparable by semi-orthogonal decompositions and we completely describe all components appearing in these semi-orthogonal decompositions. We show how this general framework encompasses many well-known semi-orthogonal decompositions. We then proceed…

## 133 Citations

Geometric invariant theory and derived categories of coherent sheaves

- Mathematics
- 2013

Given a quasiprojective algebraic variety with a reductive group action, we describe a relationship between its equivariant derived category and the derived category of its geometric invariant theory…

Combinatorial constructions of derived equivalences

- Mathematics
- 2016

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between…

REMARKS ON Θ-STRATIFICATIONS AND DERIVED CATEGORIES

- Mathematics
- 2015

This note extends some recent results on the derived category of a geometric invariant theory quotient to the setting of derived algebraic geometry. Our main result is a structure theorem for the…

Fe b 20 16 COMBINATORIAL CONSTRUCTIONS OF DERIVED EQUIVALENCES

- Mathematics
- 2021

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of Špenko and Van den Bergh, we construct equivalences between the…

Θ-STRATIFICATIONS, Θ-REDUCTIVE STACKS, AND APPLICATIONS

- Mathematics
- 2016

These are expanded notes on a lecture of the same title at the 2015 AMS summer institute in algebraic geometry. We give an introduction and overview of the “beyond geometric invariant theory” program…

The homological projective dual of Sym^2 P(V)

- Mathematics
- 2015

We study the derived category of a complete intersection X of bilinear divisors in the orbifold Sym^2 P(V). Our results are in the spirit of Kuznetsov's theory of homological projective duality, and…

On the derived categories of degree d hypersurface fibrations

- Mathematics
- 2014

We provide descriptions of the derived categories of degree d hypersurface fibrations which generalize a result of Kuznetsov for quadric fibrations and give a relative version of a well-known theorem…

A CATEGORY OF KERNELS FOR GRADED MATRIX FACTORIZATIONS AND ITS IMPLICATIONS FOR HODGE THEORY

- Mathematics
- 2012

We provide a matrix factorization model for the continuous internal Hom, in the homotopy category of k-linear dg-categories, between dg-categories of graded matrix factorizations. This motivates a…

## References

SHOWING 1-10 OF 133 REFERENCES

Variation of geometric invariant theory quotients

- Mathematics
- 1994

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of…

Geometric invariant theory and flips

- Mathematics
- 1994

We study the dependence of geometric invariant theory quotients on the choice of a linearization. We show that, in good cases, two such quotients are related by a flip in the sense of Mori, and…

From exceptional collections to motivic decompositions

- Mathematics
- 2012

Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X) of…

Non-Birational Twisted Derived Equivalences in Abelian GLSMs

- Mathematics
- 2010

In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions. Examples…

A CATEGORY OF KERNELS FOR GRADED MATRIX FACTORIZATIONS AND ITS IMPLICATIONS FOR HODGE THEORY

- Mathematics
- 2012

We provide a matrix factorization model for the continuous internal Hom, in the homotopy category of k-linear dg-categories, between dg-categories of graded matrix factorizations. This motivates a…

The derived category of a GIT quotient

- Mathematics
- 2012

Given a quasiprojective algebraic variety with a reductive group action, we describe a relationship between its equivariant derived category and the derived category of its geometric invariant theory…

Triangulated categories of singularities and D-branes in Landau-Ginzburg models

- Mathematics
- 2003

In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a…

A geometric approach to Orlov’s theorem

- MathematicsCompositio Mathematica
- 2012

Abstract A famous theorem of D. Orlov describes the derived bounded category of coherent sheaves on projective hypersurfaces in terms of an algebraic construction called graded matrix factorizations.…

Mirror symmetry for weighted projective planes and their noncommutative deformations

- Mathematics
- 2004

We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror…

The quantization conjecture revisited

- Mathematics
- 1998

A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X, L), the cohomologies of L over the GIT…