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V. Tosatti - $C^{1,1}$ estimates for complex Monge-Ampère equations

I will discuss a method that we recently introduced in collaboration with Chu and Weinkove which gives interior C1,1 estimates for the non-degenerate complex Monge-Ampère equation on compact Kähler manifolds (possibly with boundary). The method is sufficiently robust to also give C1,1 regularity of geodesic segments in the space of Kähler metrics (thus resolving a long-standing problem originating from the work of Chen), of quasi-psh envelopes in Kähler as well as nef and big classes (solving a conjecture of Berman), and of geodesic rays that arise ...
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