Variation of geometric invariant theory quotients and derived categories

Details Variation of geometric invariant theory quotients and derived categories Variation of geometric invariant theory quotients and derived categories

2012-03-29

arxiv.org/abs/1203.6643v2

Mathematics

Algebraic Geometry

91 pages, 2 figures, comments are encouraged!; v2 slight change to the introduction

Scientific paper

We develop a framework for studying the relationship between bounded derived categories of coherent sheaves on smooth global quotient stacks related by variations of the linearization in geometric invariant theory. We extend this framework to cover derived categories of coherent (matrix) factorizations when the stacks are equipped with potentials. Under assumptions on the variation, we provide simple numerical conditions for the derived categories to be related by semi-orthogonal decompositions. We also describe the complementary components in these semi-orthogonal decompositions. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on smooth and projective toric Deligne-Mumford stacks. We also show how the semi-orthogonal decompositions for derived categories of coherent factorizations fully generalize the commutative case of Orlov's $\sigma$-model/Landau-Ginzburg theorem. In addition, we present examples to show close ties with Homological Projective Duality and examples giving new derived equivalences between birational varieties. We also define a notion of a Morse-Smale action and show birational maps resulting from Morse-Smale actions admit categorical strong factorizations.

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