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Singular Kähler-Einstein metrics. (English) Zbl 1215.32017
The paper deals with the complex Monge-Ampère equation on compact Kähler manifolds, but with the background form which is not Kähler. Let \(X\) be an \(n\)-dimensional compact Kähler manifold with a semi-positive big \((1,1)\) form \(\omega\) such that \(\int _X \omega^n >0\). For a suitably normalized \(f\geq0 , \;f\in L^p (X), \;p>1\), there exists a bounded (and, under an additional technical assumption, continuous) solution \(u\) of the complex Monge-Ampère equation
\[ (\omega +dd^c u)^n =f\omega ^n, \]
for which \(\omega +dd^c u\) is a non-negative current. For \(\omega\) a Kähler form, the result is due to the reviewer [Acta Math. 180, No. 1, 69–117 (1998; Zbl 0913.35043)].
The statement is then used for \(X\) projective and \(\omega\) which is Kähler outside an analytic set \(S\) and satisfies \(\omega ^n = D\Omega ^n\) for a Kähler form \(\Omega\) and \(D^{-a} \in L^1 (\Omega ),\;a>0,\) \( [\omega ], [\Omega ] \in NS_{\mathbb R } (X).\) If \(\sigma _j , \tau _j\) are collections of holomorphic sections of two line bundles with
\[ \int_X \frac{1}{\sum |\tau _j |^{2l} }\Omega ^n <\infty \]
and
\[ \int _X \frac{\sum |\sigma _j |^{2k}}{\sum |\tau _j |^{2l} } e^F \Omega ^n=\int_X \omega^n, \]
for nonnegative numbers \(k,l\) and smooth \(F\), then the Monge-Ampère equation with right hand side
\[ \frac{\sum |\sigma _j |^{2k}}{\sum |\tau _j |^{2l} } e^F \Omega ^n \]
has a bounded solution which is smooth away from \(S\) and zero sets of the sections. This is a version of Theorem 8 from S.-T. Yau’s paper [Commun. Pure Appl. Math. 31, 339–411 (1978; Zbl 0369.53059)] with a stronger hypothesis on the manifold but weaker on the sections. The above solutions are then used to construct Kähler-Einstein metrics with singularities and with negative curvature on projective klt pairs. In particular, such metrics are obtained on canonical models of algebraic varieties of general type. This result can be viewed as a generalization of the constructions given by H. Tsuji [Math. Ann. 281, No. 1, 123–133 (1988; Zbl 0631.53051)] and G. Tian and Z. Zhang [Chin. Ann. Math., Ser. B 27, No. 2, 179–192 (2006; Zbl 1102.53047)].

MSC:
32W20 Complex Monge-Ampère operators
32Q20 Kähler-Einstein manifolds
32J27 Compact Kähler manifolds: generalizations, classification
14J17 Singularities of surfaces or higher-dimensional varieties
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