Wednesday April 23rd, 2025 | |
| Time: | 1:00 PM - 2:00 PM |
| Title: | Hodge theory of hypersurface singularities |
| Speaker: | Daniel Brogan, Stony Brook University |
| Location: | Math Tower 5-127 |
| Abstract: | |
| We discuss the connection between Hodge theory and singularities of complex projective algebraic varieties. First, we consider the variation of Hodge structure associated with the complete linear system of hypersurfaces in projective space of some fixed degree. This extends to a complex of pure Hodge modules. Using Nori’s connectivity theorem, we compute the cohomology sheaves of this Hodge module, yielding various geometric corollaries. Second, we discuss secant varieties and secant bundles. Using a version of the decomposition theorem for semismall maps, we compute the intersection cohomology of various secant varieties for smooth curves in projective space embedded via a sufficiently positive line bundle. Finally, we focus on secant varieties of rational normal curves. These are defined by the vanishing of minors of a certain Hankel matrix. We prove that these secant varieties are rational homology manifolds. We also compute the nearby and vanishing cycle sheaves for the determinant of a generic Hankel matrix. | |