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. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / April 11, 2018. / Chern classes, determinantal variety, equivariant Chern classes, local Euler obstruction / Includes bibliographical references. / Paolo Aluffi, Professor Directing Dissertation; Jorge Piekarewicz, University Representative; Ettore Aldrovandi, Committee Member; Kate Petersen, Committee Member; Mark van Hoeij, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_653538 |
| Contributors | Zhang, Xiping (author), Aluffi, Paolo, 1960- (professor directing dissertation), Piekarewicz, Jorge (university representative), Aldrovandi, Ettore (committee member), Petersen, Kathleen L. (committee member), Hoeij, Mark van (committee member), Florida State University (degree granting institution), College of Arts and Sciences (degree granting college), Department of Mathematics (degree granting departmentdgg) |
| Publisher | Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text, master thesis |
| Format | 1 online resource (106 pages), computer, application/pdf |
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