Remarks on the Chern classes of Calabi-Yau moduli

TL;DR

This paper proves that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, equipped with certain metrics, accurately represents the first Chern class of the extended tangent bundle in the compactified moduli space.

Contribution

It establishes a precise geometric relationship between the first Chern form and the first Chern class of canonical extensions in Calabi-Yau moduli spaces.

Findings

First Chern form equals the first Chern class of canonical extensions.

Results hold for both Hodge and Weil-Petersson metrics.

Provides a link between differential geometry and algebraic geometry in Calabi-Yau moduli.

Abstract

We prove that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors.