Benjamin Nill: 'An update on Gorenstein polytopes : reducibility and local h*-polynomials'

Abstract: Reflexive polytopes and more generally Gorenstein polytopes are the key objects in the Batyrev-Borisov mirror-symmetry construction of Calabi-Yau manifolds in Gorenstein toric Fano varieties. Moreover, they are beautiful combinatorial objects that appear prominently in the Ehrhart theory of lattice polytopes. In this talk I plan to present two recent combinatorial results regarding Gorenstein polytopes. First I will discuss a conjecture of Batyrev-Juny on Gorenstein polytopes of small Ehrhart degree corresponding to Gorenstein toric Fano varieties that have highly divisible anticanonical divisors. And second I will explain a characterization when Gorenstein polytopes are "thin", namely when their l*-polynomial vanishes. This polynomial is an Ehrhart-theoretic invariant that had been independently studied by Gelfand-Kapranov-Zelevinsky, Stanley, Karu, Borisov-Mavlyutov and Katz-Stapledon among others. I will highlight how underlying both results is a natural notion of reducibility for Gorenstein polytopes. This is partly joint work with Christopher Borger, Andreas Kretschmer, and Jan Schepers.