We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact Kähler manifolds have the optimal C 1,1 regularity on a Zariski open set. This also proves regularity of certain pluricomplex Green's functions on Kähler manifolds. We then go on to prove the same regularity for envelopes when the manifold is assumed to have boundary. As an application, we answer affirmatively a question of Ross-Witt-Nyström concerning the Hele-Shaw flow on an arbitrary Reimann surface.

We give new proofs of two implications in the Donaldson-Uhlenbeck-Yau theorem. Our proofs are based on geodesic rays of Hermitian metrics, inspired by recent work on the Yau-Tian-Donaldson conjecture.

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We show that if a Fano manifold does not admit Kahler-Einstein metrics then the Kahler potentials along the continuity method subconverge to a function with analytic singularities along a subvariety which solves the homogeneous complex Monge-Ampere equation on its complement, confirming an expectation of Tian-Yau. Comment: EpiGA Volume 3 (2019), Article Nr. 9

Let X be a compact Kähler manifold. We study plurisupported currents on X, i.e. closed, positive (1, 1)-currents which are supported on a pluripolar set. In particular, we are able present a technical generalization of Witt-Nyström's proof of the BDPP conjecture on projective manifolds, showing that this conjecture holds on X admitting at least one plurisupported current T such that [T ] is Kähler.One of the steps in our proof is to show an upper-bound for the pluripolar mass of certain envelopes of quasi-psh functions when the cohomology class is shifted, a result of independent interest. Using this, we are able to generalize an inequality of McKinnon and Roth to arbitrary pseudoeffective classes on compact Kähler manifolds.