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) K\"ahler metrics with torus symmetry. We also discuss possible extensions of this result to more general settings.