Complete Calabi-Yau metrics from smoothing Calabi-Yau complete intersections

TL;DR

The paper develops a gluing-based construction of complete Calabi–Yau metrics on the non-compact total space ${\mathcal{X}}$ that smooths a Calabi–Yau cone $V_0$, producing metrics with tangent cone at infinity $(\mathbb{C}\times V_0, \sqrt{-1}\partial\overline{\partial}|z|^2+\omega_0)$. It introduces model ends $X_0$ and $X_{p'}$, builds an approximate end metric by gluing, and uses weighted Hölder spaces to control decay and invert the end Laplacians; a fixed-point argument then perturbs to a global Ricci-flat metric via Tian–Yau–Hein theory. The analysis handles varying fiber complex structures and potential singularities by a smooth-at-infinity condition and a careful comparison to model spaces, extending previous results to generic complete intersections. The result provides a new systematic method to produce complete Calabi–Yau metrics on non-compact total spaces with maximal volume growth and prescribed tangent cones, broadening the class of known non-compact Calabi–Yau geometries and informing the study of asymptotic geometry in Calabi–Yau settings.

Abstract

We construct complete Calabi-Yau metrics on non-compact manifolds that are smoothings of an initial complete intersection $V_0$ that is a Calabi-Yau cone, extending the work of Székelyhidi (2019). The constructed Calabi-Yau manifold has tangent cone at infinity given by $\mathbb{C} \times V_0$. This construction produces Calabi-Yau metrics with fibers having varying complex structures and possibly isolated singularities.

Theorems & Definitions (19)

• Theorem 1.1
• Lemma 2.1
• proof
• Definition 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• Proposition 3.1
• proof
• Proposition 4.1: Extension of functions