Wednesday, 09 January 2019
Christian Haase: Hodge Numbers & Lattice Points
The study of toric varieties is a wonderful part of algebraic geometry that has deep connections with polyhedral geometry. There are elegant theorems, unexpected applications, and marvelous examples. [Cox, Little, Schenck]This talk is an introduction to this rich subject. As an example of the wonderfulness claimed above, we will look at generalizations of the celebrated Bernstein–Kushnirenko Theorem. It expresses the number of common zeros of \(n\) polynomials in \((\mathbb{C}^*)^n\) as the mixed volume of a polytope – a convex-geometric invariant.
If we have only \(k < n\) equations, the set of solutions will no longer be finite. However, there are still formulas relating cohomological invariants of the zero set to lattice point counts in Minkowski sums of polytopes.
This is based on joint projects with S. Di Rocco, M. Juhnke-Kubitzke, B. Nill, R. Sanyal, and T. Theobald.