Preserving Hodge Vectors of Lattice Polytopes

Abstract

Given lattice polytopes $P_1,\dots ,P_k$ contained in a $k$-dimensional subspace $U\subseteq {\mathbb{R}}^d$ and a $d$-dimensional lattice polytope $Q\subset {\mathbb{R}}^d$, we compute the Hodge vector of the Cayley polytope $P_1\ast \cdots \ast P_k\ast Q$, and show that it equals the mixed volume of $P_1,\dots ,P_k$ times the Hodge vector of the projection of $Q$ along $U$. Here, the Hodge vector of a lattice polytope is its local $h^{\ast }$-vector with leading and trailing zeroes removed. This result allows finding infinitely many high-dimensional lattice polytopes with the same Hodge vector that are not free joins. The proof relies on a closed formula for the Hodge-Deligne polynomial of generic complete intersections in the torus in terms of the bivariate/mixed $h^{\ast }$-polynomial. A special case of our construction is what we call Lawrence twists: extending the Gale transform by centrally-symmetric pairs of vectors. As applications, we can produce many new thin polytopes answering a question by Borger, Kretschmer and the second author, and we provide an alternative explanation of the thinness of $B_k$-polytopes answering a question of Selyanin.

15 pages