Some invariants of nonsingular projective varieties and complete local rings

Abstract

In this thesis, we establish some results concerning invariants of nonsingular projective varieties and complete local rings (in characteristic zero) which are defined using local cohomology and de Rham cohomology. We first study Lyubeznik numbers, invariants of local rings with coefficient fields defined using iterated local cohomology. If V is a nonsingular projective variety defined over a field of characteristic zero, we prove that the Lyubeznik numbers of the local ring at the vertex of the affine cone over V (viewing V as a closed subvariety of some projective space) are independent of the chosen embedding into projective space, by expressing these numbers in terms of the dimensions of the algebraic de Rham cohomology spaces of V. We next consider Matlis duality. We give an equivalent definition of the Matlis dual over a local ring with coefficient field k in terms of certain k-linear maps, which we call Sigma-continuous maps. We use this definition to develop a theory of Matlis duality for D-modules over formal power series rings in characteristic zero. If R = k[[x_1,..., x_n]] is such a ring, and D is the ring of k-linear differential operators on R, we show that the Matlis dual D(M) of any left D-module M can again be given a structure of left D-module; and if M is holonomic, the de Rham cohomology spaces of D(M) are k-dual to those of M. Finally, we examine the Hodge-de Rham spectral sequences associated with Hartshorne's algebraic de Rham homology and cohomology theories for a complete local ring A with a coefficient field k of characteristic zero. A priori, these objects depend on a choice of k-algebra surjection from a formal power series ring to A. We prove that, beginning with their E_2-terms, these spectral sequences depend only on A (and possibly the choice of coefficient field) and consist of finite-dimensional k-spaces, thus producing another set of numerical invariants of A. What is more, using our results on Matlis duality, we conclude that the E_2-objects in the homology and cohomology spectral sequences are k-dual to each other; whether this duality holds (as we conjecture) for the rest of the spectral sequences remains open.