Hodge Numbers from Picard-Fuchs Equations

TL;DR

This paper presents a method to compute Hodge numbers from Picard-Fuchs equations for variations of Hodge structure over the projective line, with applications to elliptic curves, K3 surfaces, and Calabi-Yau threefolds.

Contribution

It introduces a novel approach linking local exponents of Picard-Fuchs equations to the degrees of Hodge bundles, enabling computation of Hodge numbers in various geometric families.

Findings

Computed Hodge numbers for elliptic curves, K3 surfaces, and Calabi-Yau threefolds.

Established a method to determine degrees of Hodge bundles from local exponents.

Connected local differential equation data to global geometric invariants.

Abstract

Given a variation of Hodge structure over $\mathbb{P}^1$ with Hodge numbers $(1,1,\dots,1)$, we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin-Kontsevich-M\"oller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows us to compute the Hodge numbers of Zucker's Hodge structure on the corresponding parabolic cohomology groups. We also apply this to families of elliptic curves, K3 surfaces and Calabi-Yau threefolds.