arXiv research

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

Defines and studies solutions to complex equations on Hermitian manifolds.

The paper proves a comparison principle for complex Monge-Ampère flows and solves a uniqueness problem.

We compare various notions of weak subsolutions to degenerate complex Monge-Amp{è}re flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential subsolutions. Dedicated to Duong Hong Phong on the occasion of his 65th birthday.

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…

We discuss pluripotential aspects of the Monge-Ampère equations on compact Hermitian manifolds and prove $L^{\infty}$ estimates for any metric, as well as the existence of weak solutions under an extra assumption.

We develop an alternative approach to Degenerate complex Monge-Ampère equations on compact Kähler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones. We generalize to the Kähler case a theorem due to Dinew and Zhang in the projective ca…

Proves existence and uniqueness of weak solutions for specific equations.

Researchers find explicit solutions to complex Monge-Ampère equation.

We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…

Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Ampère type arising in Kähler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of …

Synthetic approach to pluripotential theory measures finite energy.

Study Hessian equations on compact Kähler manifolds with prescribed singularities.

This is a survey of some of the recent developments in the theory of complex Monge-Ampere equations. The topics discussed include refinements and simplifications of classical a priori estimates, methods from pluripotential theory, variational methods for big cohomology classes, semiclassical constructions of solutions …

We make a systematic study of (quasi-)plurisubharmonic envelopes on compact Kähler manifolds, as well as on domains of $\mathbb{C}^n$, by using and extending an approximation process due to Berman [Ber13]. We show that the quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution of a given co…

We propose a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry. The list includes both local questions as well as issues related to the compact complex manifold setting.

Smoothness proven for conical Calabi-Yau potentials on Fano cones.

New proof for stability estimates in complex equations without pluripotential theory.

Let (X,L) be a (semi-) polarized complex projective variety and T a real torus acting holomorphically on X with moment polytope P. Given a probability density g on P we introduce a new type of Monge-Ampere measure on X, defined for singular T-invariant metrics on the line bundle L, generalizing the ordinary Monge-Amper…

We extend profound results in pluripotential theory on Kahler manifolds to Sasaki setting via its transverse Kahler structure. As in Kahler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature (cscs) in terms of properness of K-energy. One main result i…

We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of Kähler metrics with constant scalar curvature (csck) on compact Kähler manifolds. Following their strategy, we prove that given a Sasaki structure (with Ree…

We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a …

The paper studies the asymptotic behavior of HCMA equations on ALE Kahler manifolds.

New approach proves existence of gravitating vortices on Riemann surfaces.

Defines operations in non-Archimedean metrics theory.

Lecture notes on using non-Archimedean geometry for complex variety degenerations.

Estimates for complex Monge-Ampère equations lead to insights on moduli spaces and singular metrics.

This is the second paper in a series of investigations of the pluripotential theory on Teichmüller space. The main purpose of this paper is to establish the Poisson integral formula for pluriharmonic functions on Teichmüller space which are continuous on the Bers compactification. We also observe that the Schwarz type …

Sharp $L^\infty$ estimates proved for complex Monge-Ampère equations.

Survey on metric SYZ conjecture and non-archimedean geometry.

Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.

The paper proves the existence of singular cscK metrics on smoothable varieties.

This is the first paper in a series of investigation of the pluripotential theory on Teichmüller space. The main purpose of this paper is to give an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space. We also show that Teichmüller space carries a natural stratified stru…

Study Kähler-Einstein potentials on stable varieties near singularities

We give a version of the comparison principle from pluripotential theory where the Monge-Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle

In this work, we study Monge-Ampere equations over closed Kähler manifolds with degenerated cohomology classes. Classic results and arguments in pluripotential theory are generalized a little bit to be applied to our situation.

This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kahler-Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in complex n-space. A fair amount of background…

Richberg technique adapted for nonlinear subequations.

Establishes Yau-Tian-Donaldson conjecture for weighted metrics.

Proves existence of Kähler-Einstein metrics in big cohomology classes.

Sharp inequalities for weighted log canonical thresholds derived.

The paper proves Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.

The paper finds transformation formulas for quaternionic complex structures.

Study on symmetric domains with Bergman metric properties.

The paper explores maximal destabilizers for both K-stability and Chow-stability in unstable situations.

It is shown that geodesics in the space of Kähler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for Kähler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to Kähler man…

We survey some recent developments in the direction of the Yau-Tian-Donaldson conjecture, which relates the existence of constant scalar curvature Kähler metrics to the algebro-geometric notion of K-stability. The emphasis is put on the use of pluripotential theory and the interpretation of K-stability in terms of non-…

Theory developed for complex Hessian measures on Hermitian manifolds.