A new complete Calabi-Yau metric on $\mathbb{C}^3$

TL;DR

This paper constructs a novel complete Calabi-Yau metric on ^3 with maximal volume growth, modeling collapsing Calabi-Yau threefolds near nodal points, and exhibiting unique geometric features related to singular tangent cones.

Contribution

It introduces a new complete Calabi-Yau metric on ^3 with specific singularity and asymptotic properties, expanding understanding of collapsing Calabi-Yau geometries.

Findings

The metric has maximal volume growth.

It models collapsing Calabi-Yau near nodal points.

It exhibits non-standard geometric features near singularities.

Abstract

Motivated by the study of collapsing Calabi-Yau threefolds with a Lefschetz K3 fibration, we construct a complete Calabi-Yau metric on $\mathbb{C}^3$ with maximal volume growth, which in the appropriate scale is expected to model the collapsing metric near the nodal point. This new Calabi-Yau metric has singular tangent cone at infinity, and its Riemannian geometry has certain non-standard features near the singularity of the tangent cone $\mathbb{C}^2/\mathbb{Z}_2 \times \mathbb{C}$, which are more typical of adiabatic limit problems. The proof uses an existence result in H-J. Hein's PhD thesis to perturb an asymptotic approximate solution into an actual solution, and the main difficulty lies in correcting the slowly decaying error terms.