Stability of the tangent bundle through conifold transitions

Abstract

Let X be a compact, Kähler, Calabi-Yau threefold and suppose X → 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 1,0 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.

Special Lagrangian cycles and Calabi-Yau transitions

“…This type of blow-up argument is now standard, e.g. [83,61,20]. We give the proof for the sake of completeness.…”

“…Next, near the singularities, the metrics g t are locally modelled on the Candelas-de la Ossa metrics g co,t . To be precise, given a singular point p i , there is λ > 0 such that the following estimate (stated as Lemma 6.6 in [20], which can be derived from the estimates in [34] and [16]) holds on the component of X ∩ {r < 1} containing p i :…”

“…After taking a subsequence, we have z i → z 0 ∈ X with r(z 0 ) > 0. By the construction in [34], the metrics g t i converge as t i → 0 to a metric g 0 on X which is balanced and agrees with g co,0 near the singularities, and the convergence is uniform on compact sets disjoint from the singularities (see also §2, §6.1 in [20] for further details). Also, we note that Ψ 3,0 t converges back to Ω 0 as t → 0 locally uniformly.…”

“…Friedman [33] showed that X satisfies the ∂ ∂lemma. It was shown in [20] that the stability of the tangent bundle is preserved by the transition, in the sense that the tangent bundle of X admits a Hermitian-Yang-Mills connection with respect to the balanced metric ω. The stability of other bundles satisfying a local triviality condition near the singular points was studied in [16].…”

“…Let Y → Y X t be a conifold transition contracting holomorphic 2-spheres C i ⊂ Y and replacing them with vanishing 3-spheres L i,t ⊂ X t . To understand this topological change geometrically, Fu-Li-Yau [34] (see also [16,20] for follow-up work) construct a sequence of metrics ω a on Y with dω 2 a = 0 and Vol(C i , ω a ) → 0 as a → 0, and ω t on X t with dω 2 t = 0 with Vol(L i,t , ω t ) → 0 as t → 0. We will perturb L i,t to a special submanifold with respect to this geometry.…”