Theta divisors and permutohedra

TL;DR

The paper reveals a deep link between smooth theta divisors $Θ^n$ on principally polarized abelian varieties and the combinatorics of permutohedra via the permutohedral toric variety $X_Π^n$. It shows that the two-parameter Todd genus $Td_{s,t}$ of $Θ^n$ coincides with the $h$-polynomial of the permutohedron, and derives explicit Hodge-number formulas in terms of Eulerian numbers. A complementary duality with $X_Π^n$ is established at the level of genera and Betti numbers, while intricate relations to Tomei manifolds and Toda lattice dynamics are explored, including nontrivial cobordism-from-toric examples for large $n$. The work thus connects complex cobordism, toric geometry, and integrable-system-inspired manifolds through precise combinatorial identities involving permutohedra and Eulerian statistics, and provides explicit descriptions of the Hodge structure of $Θ^n$ in terms of classical combinatorial numbers.

Abstract

We establish an intriguing relation of the smooth theta divisor $Θ^n$ with permutohedron $Π^n$ and the corresponding toric variety $X_Π^n.$ In particular, we show that the generalised Todd genus of the theta divisor $Θ^n$ coincides with $h$-polynomial of permutohedron $Π^n$ and thus is different from the same genus of $X_Π^n$ 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.

Figures (1)

• Figure 1: Permutohedra in dimension 1,2 and 3.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 3.1
• proof