Stability of the tangent bundle through conifold transitions

TL;DR

This paper proves that the tangent bundle of Calabi-Yau threefolds undergoing conifold transitions admits Hermitian-Yang-Mills metrics for small smoothing parameters, and analyzes the metrics' behavior near vanishing cycles.

Contribution

It establishes the existence of Hermitian-Yang-Mills metrics on tangent bundles during conifold transitions and describes their asymptotic behavior near singularities.

Findings

Existence of Hermitian-Yang-Mills metrics on tangent bundles for small smoothing parameters.

Description of metric behavior near vanishing cycles as the transition parameter approaches zero.

Extension of stability results to conifold transitions in Calabi-Yau threefolds.

Abstract

Let $X$ be a compact, K\"ahler, Calabi-Yau threefold and suppose $X\mapsto \underline{X}\leadsto X_t$ , for $t\in \Delta$, is a conifold transition obtained by contracting finitely many disjoint $(-1,-1)$ curves in $X$ and then smoothing the resulting ordinary double point singularities. We show that, for $|t|\ll 1$ sufficiently small, the tangent bundle $T^{1,0}X_{t}$ admits a Hermitian-Yang-Mills metric $H_t$ with respect to the conformally balanced metrics constructed by Fu-Li-Yau. Furthermore, we describe the behavior of $H_t$ near the vanishing cycles of $X_t$ as $t\rightarrow 0$.