Theta divisors and permutohedra

TL;DR

This paper uncovers a surprising connection between smooth theta divisors and permutohedra, linking their geometric invariants and revealing new formulas for Hodge numbers and numerical relations with integrable systems.

Contribution

It establishes a novel relation between theta divisors and permutohedra, including their Todd genus and Hodge numbers, and explores connections with Tomei manifolds.

Findings

Todd genus of theta divisor equals h-polynomial of permutohedron

Hodge numbers of theta divisors expressed via Eulerian numbers

Numerical relations between theta-divisors and Tomei manifolds

Abstract

We establish an intriguing relation of the smooth theta divisor $\Theta^n$ with permutohedron $\Pi^n$ and the corresponding toric variety $X_\Pi^n.$ In particular, we show that the generalised Todd genus of the theta divisor $\Theta^n$ coincides with $h$-polynomial of permutohedron $\Pi^n$ and thus is different from the same genus of $X_\Pi^n$ only by the sign $(-1)^n.$ As an application we find all the Hodge numbers of the theta divisors in terms of the Eulerian numbers. We reveal also interesting numerical relations between theta-divisors and Tomei manifolds from the theory of the integrable Toda lattice.