Stability of the tangent bundle through conifold transitions

Let X be a compact, Kähler, Calabi-Yau threefold and suppose X 7→ X Xt , for t ∈ ∆, 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| ≪ 1 sufficiently small, the tangent bundle T Xt admits a Hermitian-Yang-Mills metric Ht with respect to the conformally balanced metrics constructed by Fu-Li-Yau. Furthermore, we describe the behavior of Ht near the vanishing cycles of Xt as t → 0.

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