7.2. Singular Ricci flat metrics [02FR]

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7.2. Singular Ricci flat metrics

Definition 7.4.

Let $V$ be a Kähler space with only log terminal singularities. $V$ is said to be $\mathbb{Q}$ -CY, iff there is some multiple $N^{\prime}$ of $index(X)$ such that $H^{0}(V,\omega^{[N^{\prime}]}_{V})=\mathbb{C}\alpha$ , where $\alpha$ is a global generator of $\omega^{[N^{\prime}]}_{V}$ .

Theorem 7.5.

Assume $V$ is a compact $\mathbb{Q}$ -CY Kähler space. Let $\Omega$ be a smooth Kähler metric on $V$ . Then there is a unique semi-Kähler current with continuous potential and adapted Monge-Ampère measure $\Omega^{\prime}=\Omega+dd^{c}\varphi$ , such that

(\Omega+dd^{c}\varphi)^{n}=Cv_{\alpha}\text{ and }\sup_{V}\varphi=-1,

where $\int_{V}\Omega^{n}=C\int_{V}(-1)^{n}v_{\alpha}$ .

Furthermore, if $V$ is projective-algebraic and $[\Omega]\in NS_{\mathbb{R}}(V)$ , then $\Omega+dd^{c}\varphi$ is smooth on $V^{reg}$ where it defines a bona fide Ricci flat metric.

Corollary 7.6.

In each cohomology class of a smooth Kähler form, there is a unique singular Ricci flat metric..

Proof.

This follows straighforwardly from Theorems 6.3, 3.6, Lemma 6.4 and Definition 7.3. ∎

Example 7.7.

A nodal quintic threefold is $\mathbb{Q}$ -CY and has not quotient singularities, so the orbifold method of [Ko] does not work.