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\(C^{1,1}\) regularity for degenerate complex Monge-Ampère equations and geodesic rays.(English)Zbl 1404.32075

The authors prove a regularity result for the solution to the Dirichlet problem for the homogeneous complex Monge-Ampère equation on a compact Kähler manifold with boundary and in a degenerate cohomology class. The precise setting is as follows. Let \(M\) be a compact Kähler manifold of dimension \(n\) with smooth boundary \(\partial M\) and let \(\omega_0\) be a smooth semipositive closed \((1,1)\)-form on \(M\). Assume further that \(\partial M\) is weakly pseudoconcave and that there exists an effective divisor \(E\subset M\) disjoint from \(\partial M\) such that \(\omega_0-\delta c_1(\mathcal O(E),h)\) is a Kähler form on \(M\), for all sufficiently small \(\delta>0\) and some Hermitian metric \(h\) on the line bundle \(\mathcal O(E)\) associated to \(E\). Let \(\varphi_0\) be a smooth function on \(M\) such that \(\omega_0+i\partial\overline\partial\varphi_0\geq0\). Let \(\varphi\) be the unique \(\omega_0\)-plurisubharmonic (psh) function such that \((\omega_0+i\partial\overline\partial\varphi)^n=0\) on \(M\), \(\varphi=\varphi_0\) on \(\partial M\), which exists by a standard envelope construction (Perron method). Under these assumptions it is shown in Theorem 1.1 that \(\varphi\in C_{\mathrm{loc}}^{1,1}(M\setminus E)\). This improves a result ofD. H. Phong andJ. Sturm who showed that \(\varphi\in C_{\mathrm{loc}}^{1,\alpha}(M\setminus E)\) for all \(0<\alpha<1\) [Commun. Anal. Geom. 18, No. 1, 145–170 (2010;Zbl 1222.32044)] (see also [S. Boucksom, Lect. Notes Math. 2038, 257–282 (2012;Zbl 1231.32025)]).
The authors apply the above result to prove the local \(C^{1,1}\) regularity of geodesic rays in the space of Kähler metrics associated to a test configuration (Theorem 1.2). They also use it to study the regularity of the \(\alpha\)-psh envelope \[u=\sup\{\varphi:\,\varphi\in \operatorname{PSH}(M,\alpha),\,\varphi\leq0\}\,,\] where \([\alpha]\) is a big and nef \((1,1)\)-class on a compact Kähler manifold \((M,\omega)\). They show that \(u\) is locally \(C^{1,1}\) away from the non-Kähler locus of \([\alpha]\), extending earlier results ofR. J. Berman [Am. J. Math. 131, No. 5, 1485–1524 (2009;Zbl 1191.32008)] andR. Berman andJ.-P. Demailly [Prog. Math. 296, 39–66 (2012;Zbl 1258.32010)].

MSC:

32W20 Complex Monge-Ampère operators
35J96 Monge-Ampère equations
32Q15 Kähler manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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