Existence of Hermitian-Yang-Mills metrics under conifold transitions

TL;DR

This paper investigates the behavior and existence of Hermitian-Yang-Mills metrics on Calabi-Yau threefolds undergoing conifold transitions, providing new existence results under degeneration conditions.

Contribution

It establishes the existence of Hermitian-Yang-Mills metrics on bundles over threefolds after conifold transitions, extending previous understanding of metric degeneration and stability.

Findings

Degeneration of Hermitian-Yang-Mills metrics analyzed on Calabi-Yau threefolds.

Existence of Hermitian-Yang-Mills metrics proved on conifold transition spaces.

Construction of metrics on degenerating and transitioned Calabi-Yau manifolds.

Abstract

We first study the degeneration of a sequence of Hermitian-Yang-Mills metrics with respect to a sequence of balanced metrics on a Calabi-Yau threefold $\hat{X}$ that degenerates to the balanced metric constructed by Fu, Li, and Yau on the complement of finitely many (-1,-1)-curves in $\hat{X}$. Then under some assumptions we show the existence of Hermitian-Yang-Mills metrics on bundles over a family of threefolds $X_t$ with trivial canonical bundles obtained by performing conifold transitions on $\hat{X}$.