| This dissertation is divided into two parts. In the first part (Chapter 1) we study deformation theory of algebraic cycles. Algebraic cycles are an all-important but still poorly-understood subject in algebraic geometry. Infinitesimal methods are introduced by Griffiths and Green [GG] to study families of cycles. We take up on their idea and investigate its geometric consequences. Although we still do not know what the deformation and obstruction spaces to algebraic cycles should be, our investigation has led to some general deformation-theoretic results.; Chapter 2 serves as an appendix to Chapter 1. It is an exposition of Voisin's result that if d ≥ 7, two distinct points on a generic degree d-hypersurface in 3 cannot be rationally equivalent. We state an analogous result which works for an arbitrary number of points.; In the second part (Chapter 3) we construct a new geometric invariant called intersection torsion associated to a large class of spaces, which include real and complex algebraic varieties. This construction generalizes Reidemeister torsion in the same way intersection cohomology is a generalization of ordinary cohomology. Although intersection torsion behaves like a topological invariant, some mild additional data is needed in its construction, such as the algebraic structure afforded by varieties. |
