Complex structure degenerations and collapsing of Calabi-Yau metrics

Abstract

In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space degenerating into the transversal union of two smooth Fano hypersurfaces in a generic way, we obtain a complete result in all dimensions establishing explicit and precise relationships between the metric collapsing and complex structure degenerations. This result is new even in complex dimension two. This is achieved via gluing and singular perturbation techniques, and a key geometric ingredient involving the construction of certain (not necessarily smooth) Kähler metrics with torus symmetry. We also discuss possible extensions of this result to more general settings.

Minor revision and updates

References in corpus (4)

• Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces
• Resolution of the canonical fiber metrics for a Lefschetz fibration
• SYZ geometry for Calabi-Yau 3-folds: Taub-NUT and Ooguri-Vafa type metrics
• A Liouville theorem on asymptotically Calabi spaces

Cited by in corpus (8)

• SYZ conjecture for Calabi-Yau hypersurfaces in the Fermat family
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• The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on ${\mathbb{T}}^4$
• Collapsing Ricci-flat metrics on elliptic K3 surfaces
• Polystable log Calabi-Yau varieties and Gravitational instantons
• Collapsing geometry of hyperkähler 4-manifolds and applications
• Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures
• Boundary asymptotics of the relative Bergman kernel metric for curves