On the degeneration of asymptotically conical Calabi-Yau metrics

TL;DR

This paper investigates how asymptotically conical Calabi-Yau metrics degenerate when the K"ahler class approaches a semi-positive limit, revealing convergence to incomplete Ricci-flat metrics and constructing singular Calabi-Yau metrics with specific asymptotic behavior.

Contribution

It demonstrates the convergence of degenerating Ricci-flat K"ahler metrics to incomplete metrics and constructs new singular Calabi-Yau metrics on quasi-projective varieties.

Findings

Ricci-flat K"ahler metrics converge to incomplete metrics away from a subvariety

Constructs singular Calabi-Yau metrics with asymptotically conical behavior

Shows the homeomorphism between metric geometry and the topology of the singular variety

Abstract

We study the degenerations of asymptotically conical Ricci-flat K\"ahler metrics as the K\"ahler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat K\"ahler metrics converge to a incomplete smooth Ricci-flat K\"ahler metric away from a compact subvariety. As a consequence, we construct singular Calabi-Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our results to study several classes of examples of geometric transitions between Calabi-Yau manifolds.