Stability of the tangent bundle through conifold transitions

Abstract

Let $X$ be a compact, Kähler, Calabi-Yau threefold and suppose $X\mapsto \underline{X}⇝X_t$ , for $t\in \hat{I}”$, 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 $t\to 0$.

80 pages, 1 appendix; v2 minor updates