Characteristic Classes and Local Invariants of Determinantal Varieties and a Formula for Equivariant Chern-Schwartz-MacPherson Classes of Hypersurfaces

Determinantal varieties parametrize spaces of matrices of given ranks. The main results of this dissertation are computations of intersection-theoretic invariants of determinantal varieties. We focus on the Chern-Mather and Chern-Schwartz-MacPherson classes, on the characteristic cycles, and on topologically motivated invariants such as the local Euler obstruction. We obtain explicit formulas in both the ordinary and the torus-equivariant setting, and formulate a conjecture concerning the effectiveness of the Chern-Schwartz-MacPherson classes of determinantal varieties. We also prove a vanishing property for the Chern-Schwartz-MacPherson classes of general group orbits. As applications we obtain formulas for the sectional Euler characteristic of determinantal varieties and the microlocal indices of their intersection cohomology sheaf complexes. Moreover, for a close embedding we define the equivariant version of the Segre class and prove an equivariant formula for the Chern-Schwartz-MacPherson classes of hypersurfaces of projective varieties.