arXiv research

Synthetic approach to pluripotential theory measures finite energy.

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.

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…

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

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

Defines operations in non-Archimedean metrics theory.

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…

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

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…

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

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…

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

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 …

Defines and studies solutions to complex equations on Hermitian manifolds.

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.

Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.

Survey on metric SYZ conjecture and non-archimedean geometry.

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

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 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

Proves existence and uniqueness of weak solutions for specific equations.

Richberg technique adapted for nonlinear subequations.

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 …

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

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.

Study on symmetric domains with Bergman metric properties.

Establishes Yau-Tian-Donaldson conjecture for weighted metrics.

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.

Sharp inequalities for weighted log canonical thresholds derived.

The purpose of this paper is to establish a Lagrangian potential theory, analogous to the classical pluripotential theory, and to define and study a Lagrangian differential operator of Monge-Ampere type. This development is new even in ${\bf C}^n$. However, it applies quite generally -- perhaps most importantly to symp…

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.

New approach proves existence of gravitating vortices on Riemann surfaces.

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

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

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 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 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-…

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 …

We give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted Kähler-Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve the continuity method or Cheeger-Colding-Tian the…

Study Hessian equations on compact Kähler manifolds with prescribed singularities.

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

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…

The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,α} estimates away from a divisor are obtained, by combining techniques of Blocki, Tsuji, Yau, and pluripotential theory. In particular, C^{1,α} geod…

The paper proves a unique cscK metric for uniformly K-stable Kähler manifolds.