Calabi-Yau metrics on rank two symmetric spaces with horospherical tangent cone at infinity

Abstract

We show that on every non-$G_2$ complex symmetric space of rank two, there are complete Calabi-Yau metrics of Euclidean volume growth with prescribed horospherical singular tangent cone at infinity, providing the first examples of affine Calabi-Yau smoothings of singular and irregular tangent cone. As a corollary, we obtain infinitely many examples of Calabi-Yau manifolds degenerating to the tangent cone in a single step, supporting a recent conjecture by Sun-Zhang, which was only proved when the tangent cone at infinity has only an isolated singularity.

45 pages. Section 2 was substantially reworked and more details on the geometry of the compactification was provided. A gap in the proof of Proposition 7.2 was fixed by strengthening Proposition 7.1. An appendix on spherical varieties was added at the end. To appear in J. Differ. Geom