7.4. Singular KE metrics on klt pairs [02G3]

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7.4. Singular KE metrics on klt pairs

Let us now state the immediate generalization to klt pairs.

Definition 7.11.

Let $(V,\Delta)$ be a klt compact Kähler pair.

The pair $(V,\Delta)$ is said to be $\mathbb{Q}$ -CY, iff there is some multiple $N^{\prime}$ of $index(X,\Delta)$ such that $H^{0}(V,\mathcal{O}_{V}(N^{\prime}(K_{V}+\Delta)))=\mathbb{C}\alpha$ where $\alpha$ is a global generator of $\mathcal{O}_{V}(N^{\prime}(K_{V}+\Delta))$ .

The pair $(V,\Delta)$ is canonically polarized iff $K_{V}+\Delta$ is ample.

Theorem 7.12.

Let $(V,\Delta)$ be a klt compact Kähler pair.

If $(V,\Delta)$ is $\mathbb{Q}$ -CY it carries a singular Ricci flat metric with adapted volume form in any Kähler class of $V$ , this current being smooth outside $\Delta\cup V^{sing}$ if $V$ projective and the Kähler class is rational.

If it is canonically polarized it carries a unique singular KE metric in the cohomology class of $K_{V}+\Delta$ , regular outside $\Delta\cup V^{sing}$ .

Proof.

For regularity on the smooth locus, we need the full statement of Theorems 3.5 and 4.5, poles included. ∎

Acknowledgements. We would like to thank Z. Blocki, S. Boucksom, A. Chiodo, J. Keller, S. Kolodziej, M. Paun and B. Toën for useful conversations and C. Simpson and Y.T. Siu for inspiring remarks.