7. Singular Kähler-Einstein metrics

7.1. Singular Ricci curvature

The smooth case

The link between Monge-Ampère equations and Kähler-Einstein metrics is provided by the following classical

Lemma 7.1.

Let $X$ be complex manifold, let $h$ be a smooth hermitian metric on $\omega_{X}$ and $\Omega$ a Kähler form such that $\Omega^{n}=v(h)$ . The Ricci curvature divided by $2\pi$ of $\Omega$ is the Chern-Weil form $-c_{1}(K_{X},h)$ .

Adapted measures and hermitian metrics on the canonical sheaf

Assume $V$ is compact with only log terminal singularities, has index $N$ and let $h^{N}$ be a smooth hermitian metric on $\omega_{V}^{[N]}$ . Let $\beta$ be a local generator local of $\omega_{V}^{[N]}$ . Define $v_{\beta}(h)$ to be the volume form on $V^{reg}$ :

v_{\beta}(h)=\left(\sqrt{-1}^{Nn}(-1)^{N\frac{n(n+1)}{2}}\frac{\beta\wedge\bar{\beta}}{\|\beta\|^{2}_{h^{N}}}\right)^{\frac{1}{N}}

Since $v_{\beta}(h)$ is independent of $\beta$ , this expression defines an adapted measure $v(h)$ with ${\mathcal{C}}^{\infty}$ density on $V$ .

Now, let $h^{N}_{sing}=e^{-N\chi}h^{N}$ be a singular metric on $\omega_{V}^{[N]}$ . The Chern-Weil form $c_{1}(\omega_{X}^{N},h_{sing}^{N})$ is then well defined as a quasipositive current. Since $h_{sing}^{N}$ has locally ${\mathcal{C}}^{\infty}$ +psh potentials $\chi$ is locally bounded above and the above formula defines a measure $v(h_{sing})=e^{\chi}v(h)$ on $V$ such that $\frac{v(h_{sing})}{v(h)}\in L^{\infty}_{loc}$ . In particular

\frac{v(h_{sing})}{\Omega^{n}}\in L^{1+\varepsilon}(V,\Omega^{n})\text{ for }\varepsilon>0\text{ small enough}.

We have $c_{1}(K_{X},h_{sing})=c_{1}(K_{X},h)+dd^{c}\chi$ , where $c_{1}(K_{X},h):=\frac{1}{N}c_{1}(\omega^{N}_{X},h^{N})$ .

Definition 7.2.

Assume $V$ has only log terminal singularities. An adapted measure on $V$ is a positive Radon measure locally of the form $e^{f}.v$ where $f$ is locally given as the sum of a psh and a smooth function on $V$ . An adapted measure is ${\mathcal{C}}^{0}$ , ${\mathcal{C}}^{\alpha}$ , ${\mathcal{C}}^{\infty}$ density if so is $e^{f}$ .

The definition has the virtue of generalizing the usual equivalence between smooth metrics on the canonical sheaf of a manifold and positive definite volume forms to singular metrics and log terminal spaces. This suggests the following ad hoc:

Definition 7.3.

Let $V$ be a $\mathbb{Q}$ -Gorenstein Kähler normal $n$ -dimensional complex space with only log terminal singularities. Let $\Omega$ be a semi-Kähler current with $L^{\infty}_{loc}$ potential and adapted Monge-Ampère measure. Let $h$ be the singular metric on the canonical sheaf such that $\Omega^{n}=v(h)$ . We define

Ric(\Omega):=-c_{1}(K_{V},h),

where the equality is to be taken in the sense of currents.

The metric $\Omega$ will be called a singular Kähler-Einstein metric if $Ric(\Omega)=c\Omega$ for some $c\in\mathbb{R}$ .

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.

7.3. Singular Kähler-Einstein metrics of negative curvature

Theorem 7.8.

Let $V$ be a general type projective algebraic variety with only canonical singularities such that $K_{V}$ is ample. Let $h^{N}$ a smooth hermitian metric on $\omega^{N}_{V}$ such that $\Omega=c_{1}(K_{V},h)$ is a smooth Kähler form on $V$ .

There is a unique $\varphi\in{\mathcal{C}}^{0}(V,\mathbb{R})$ such that:

(1)

$\varphi$ is $\Omega$ -psh.
(2)

$\Omega+dd^{c}\varphi$ semi Kähler current with ${\mathcal{C}}^{0}$ potential.
(3)

$(\Omega+dd^{c}\varphi)^{n}=e^{\varphi}v(h)$ .

Consequently $\Omega+dd^{c}\varphi$ is the unique singular KE metric on $V$ of negative curvature in the canonical class of $V$ . The current $\Omega+dd^{c}\varphi$ has continuous potentials and is smooth on $V^{reg}$ where it defines a bona fide KE metric.

Proof.

This is a consequence of Theorems 4.1, 4.4, and Definition 7.3. ∎

Remark 7.9.

Thanks to Theorem 5.4, for $X$ a projective algebraic manifold of general type such that $R(X):=\oplus_{n\in\mathbb{N}}H^{0}(X,\mathcal{O}_{X}(nK_{X}))$ is finitely generated, $X$ has a unique birational model $V$ such that the above hypotheses hold. Thus we have a birational map $\pi:X\dashrightarrow V$ which is well defined outside an indeterminacy locus $S$ of codimension $\leq 2$ . In particular $\pi^{*}(\omega+dd^{c}\varphi)$ is a closed positive current on $X-S$ that extends to a closed positive current $T$ on $X$ itself. The current $T$ defines a KE metric on $X-S$ . It needs not be a singular KE metric on $X$ though, since its potentials may have logarithmic poles on $S$ , in fact algebraic singularities of the form $\alpha\log(\sum|f_{i}|^{2})+O(1)$ $f_{i}$ holomorphic and $\alpha\in\mathbb{Q}_{>0}$ . Moreover, $T$ lies in the canonical class of $X$ iff $X$ is a smooth minimal model as in [Ts].

Connection with [Ts]

Let $X$ be a complex projective manifold such that $K_{X}$ is nef and big. Let $\Omega$ be a smooth Kähler metric on $X$ and consider the Kähler-Ricci flow

\frac{\partial\Omega_{t}}{\partial t}=-Ric(\Omega_{t})-\Omega_{t},\ \ \Omega_{0}=\Omega.

In [Ts], it was proved that this flow has a global solution for all time $t\in[0,\infty[$ , and an argument was given, recently fully completed in [TZ], to the effect that $\Omega_{t}$ converges to a closed positive current $T_{KE}$ , independent of $\Omega$ , which defines a smooth Kähler-Einstein metric outside the exceptional divisor $E$ of the holomorphic bimeromorphic map $X\to X_{can}$ . Its potential satisfies the Monge Ampère équation considered in Theorem 7.8 outside $E$ . It follows from proposition 4.4 that the current $T_{KE}$ coincides with the solution produced by Theorem 7.8.

The notes [ST], [TZ] announce a proof of the following properties, already conjectured by [Ts], that $T_{KE}$ has locally bounded potential and satisfies the degenerate Monge-Ampère equation considered in Theorem 7.8. Our Theorem 7.8 in this case gives the precision that $T_{KE}$ has continuous potentials.

Example 7.10.

A nodal sextic threefold is of general type, Gorenstein, terminal, is its own canonical model, has no smooth minimal model and does not have quotient singularities. Therefore the orbifold method of [Ko] does not work and [Ts] does not apply.

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.

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7. Singular Kähler-Einstein metrics [02FJ]