Degenaration of Calabi-Yau Manifold with W-P Metric

TL;DR

This paper characterizes when degenerations of Calabi-Yau manifolds are at finite Weil-Petersson distance, providing cohomological conditions and implications for physics phenomena like massless black holes.

Contribution

It introduces a simple cohomological criterion for finite distance degenerations of Calabi-Yau manifolds based on mixed Hodge structures.

Findings

Central fibre with simple nodes is at finite Weil-Petersson distance.

A cohomological condition determines finite distance degenerations.

The results connect geometric degenerations with physical concepts like massless black holes.

Abstract

In this paper we focus on determining for which degenerations the central fibre is at finite distance with respect to Weil-Petersson metric. We obtain a simple condition on the limiting mixed Hodge structure. Then we combine the result with the canonical mixed Hodge structure of the central fibre and obtain a simple cohomological condition for the central fibre to be at finite distance. As a corollary, we prove that a central fibre with simple nodes is at finite distance. This issue has been raised in the Physics literature including the recent development of so-called "mass-less black holes".