Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities

Abstract

Let $X_0$ be an affine variety with only normal isolated singularity $p$ and $\pi :X\to X_0$ a smooth resolution of the singularity with trivial canonical line bundle $K_X$. If the complement of the affine variety $X_0\backslash \{p\}$ is the cone $C(S)={\mathbb{R}}_{>0}\times S$ of an Einstein-Sasakian manifold $S$, we shall prove that the crepant resolution $X$ of $X_0$ admits a complete Ricci-flat Kähler metric in every Kähler class in $H^2(X)$. We apply the continuity method for solving the Monge-Ampère equation to obtain a relevant existence theorem and a uniqueness theorem of Ricci-flat conical Kähler metrics. By using the vanishing theorem on the crepant resolution $X$ and the Hodge and Lefschetz decompositions of the basic cohomology groups on the Sasakian manifold $S$, we construct an initial Kähler metric in every Kähler class on which the existence theorem can be applied.We show there are many examples of Ricci-flat complete Kähler manifolds arising as crepant resolutions.

arXiv commentLemma 5.3 was improved. The transverse Dolbeault theorem is not necessary to show the vanishing of the cohomology groups of Lemma 5.3. Theorem 1.8 (uniqueness theorem) was also improved

References in corpus (2)

• Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds
• Regularity of asymptotically conical Ricci-flat Kähler metrics

Cited by in corpus (2)

• Baryonic symmetries and M5 branes in the AdS_4/CFT_3 correspondence
• Regularity of asymptotically conical Ricci-flat Kähler metrics