Mixed Hodge Structures on Generalized Theta Divisors and Graph Motives

TL;DR

This paper explores the mixed Hodge structures on the complements of generalized theta divisors in generalized Jacobians of curves, providing explicit formulas and linking them to graph motives and Feynman integrals.

Contribution

It introduces explicit formulas for the weight filtration on cohomology and suggests a conjectural connection to graph motives related to Feynman integrals.

Findings

Explicit formulas for the weight filtration on cohomology.

Examples include nodal cubic and low genus singular curves.

Connections to graph motives and Feynman integrals are proposed.

Abstract

This article studies the mixed Hodge structures that appear on the complements of generalized theta divisors inside generalized Jacobians of curves with modulus. For a smooth or nodal curve with an effective modulus, the generalized Jacobian is a semiabelian variety, and its generalized theta divisor has a natural determinantal description. Using a smooth compactification with simple normal crossing boundary together with Deligne's theory of logarithmic mixed Hodge complexes, we obtain explicit formulas for the weight filtration on the cohomology of the complement of the generalized theta divisor. The graded pieces of the weight filtration are described in terms of the cohomology of strata determined by the dual graph of the curve and the combinatorics of the modulus. Several examples are worked out, including the nodal cubic and low genus singular curves, showing cases of mixed Tate type as well as more complicated weight behavior. Motivated by similarities with the structure of graph hypersurfaces, we also suggest a conjectural connection between these mixed Hodge structures and motives associated to decorated dual graphs related to Feynman integrals.