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arXiv · 1705.07026

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

Yang Li
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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.

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Yang Li. 2017-05-19. A new complete Calabi-Yau metric on $\mathbb{C}^3$. https://arxiv.org/abs/1705.07026

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