arXiv research

Introduces trace operator for quasi-plurisubharmonic functions on Kähler manifolds.

Continuous solutions found for complex geometry equations.

Researchers solve Dirichlet problem for complex Monge-Ampère equation on Hermitian manifolds.

New stability thresholds detect K-stability in Fano manifolds.

New method for complex Monge-Ampère equations on Kähler manifolds.

Study on finite entropy and energy in Kähler geometry.

We introduce a wide subclass ${\cal F}(X,ω)$ of quasi-plurisubharmonic functions in a compact Kähler manifold, on which the complex Monge-Ampère operator is well-defined and the convergence theorem is valid. We also prove that ${\cal F}(X,ω)$ is a convex cone and includes all quasi-plurisubharmonic functions which are …

New method solves complex Monge-Ampère equations on hermitian manifolds.

Study bounds on Monge-Ampère volumes for degenerate complex equations.

Existence and uniqueness of bounded solutions to complex Monge-Ampère flows on Kähler manifolds.

We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.

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…

Let $X$ be a compact Kähler manifold and $θ$ a smooth closed $(1,1)$-real form representing a big cohomology class $α\in H^{1,1}(X,\R)$. The purpose of this note is to show, using pluripotential and viscosity techniques, that any $θ$-plurisubharmonic function $\f$ can be approximated from above by a decreasing sequence…

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answ…

In this article, we solve the strong openness conjecture on the multiplier ideal sheaves for the plurisubharmonic functions posed by Demailly. We prove two conjectures about the growth of the volumes of the sublevel sets of plurisubharmonic functions related to the complex singularity exponents and quasi-plurisubharmon…

Let $X$ be a compact Kähler unibranch complex analytic space of pure dimension. Fix a big class $α$ with smooth representative $θ$ and a model potential $φ$ with positive mass. We define and the study non-pluripolar products of quasi-plurisubharmonic functions on $X$. We study the spaces $\mathcal{E}^p(X,θ;[φ])$ of fin…

The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes …

We introduce the notion of the projective linking number Link(M,Z) of a compact oriented real submanifold M of dimension 2p-1 in complex projective n-space P^n with an algebraic subvariety Z in P^n - M of codimension p. This notion is related to projective winding numbers and quasi-plurisubharmonic functions, and it ge…

Analyzes Kähler-Einstein metrics on families of Fano varieties.

Characterizes closures of test configurations and algebraic singularity types.