Geometric invariant theory and derived categories of coherent sheaves

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 quotient. This generalizes classical descriptions of the category of coherent sheaves on projective space and categorifies several results in the theory of Hamiltonian group actions on projective manifolds.This perspective generalizes and provides new insight into examples of derived equivalences between birational varieties. We provide a criterion under which two different GIT quotients are derived equivalent, and apply it to prove that any two generic GIT quotients of an equivariantly Calabi-Yau projective-over-affine variety by a torus are derived equivalent.We also use these techniques to study autoequivalences of the derived category of coherent sheaves of a variety arising from a variation of GIT quotient. We show that these autoequivalences are generalized spherical twists, and describe how they result from mutations of semiorthogonal decompositions. Beyond the GIT setting, we show that all generalized spherical twist autoequivalences of a dg-category can be obtained from mutation in this manner.Motivated by a prediction from mirror symmetry, we refine the main theorem describing the derived category of a GIT quotient. We produce additional derived autoequivalences of a GIT quotient and propose an interpretation in terms of monodromy of the quantum connection. We generalize this observation by proving a criterion under which a spherical twist autoequivalence factors into a composition of other spherical twists.Finally, our technique for studying the derived category of a GIT quotient relies on a special stratification of the unstable locus in GIT. In the final chapter we establish a new modular description of this stratification using the mapping stack from the quotient of the affine line by the multiplicative group to the quotient stack X/G. This is the first foundational step in extending the methods of GIT beyond global quotient stacks X/G to other stacks arising in algebraic geometry. We describe a method of constructing such stratifications for arbitrary algebraic stacks and show that it reproduces the GIT stratification as well as the classical stratification of the moduli stack of vector bundles on a smooth curve