C1,1 regularity for degenerate complex Monge-Amp\`ere equations and geodesic rays
Abstract
We prove a C1,1 estimate for solutions of complex Monge-Amp\`ere equations on compact K\"ahler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local C1,1 regularity of geodesic rays in the space of K\"ahler metrics associated to a test configuration, as well as the local C1,1 regularity of quasi-psh envelopes in nef and big classes away from the non-K\"ahler locus.
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