Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities (Q455016)

The main result is Theorem 5.1: Let \(X_0\) be an affine variety with only one normal isolated singularity at \(p.\) Suppose that the complement of \(p\) in \(X_0\) is biholomorphic to the cone \(C(S)\) of an Einstein-Sasakian manifold \(S\) of real dimension \(2n-1.\) If there is a resolution of singularity \(\pi:X\rightarrow X_0\) with trivial canonical line bundle \(K_X,\) then there is a Ricci-flat complete Kähler metric for every Kähler class of \(X.\)NEWLINENEWLINEThe paper also obtains a uniqueness theorem of Ricci-flat conical Kähler metrics in each Kähler class with a certain boundary condition and gives many examples of Ricci flat complete Kähler manifolds arising as crepant resolutions.NEWLINENEWLINEThe main theorem covers the crucial case where a Kähler class does not belong to the compactly supported cohomology group. The author uses a vanishing theorem and the Hodge and the Lefschetz decomposition theorems to construct a suitable initial Kähler metric in every Kähler class to which the existence theorem can be applied.