Related papers: Non-pluripolar energy and the complex Monge-Amp\`e…

Suppose that $X$ is an analytic subvariety of a Stein manifold $M$ and that $\varphi$ is a plurisubharmonic (psh) function on $X$ which is dominated by a continuous psh exhaustion function $u$ of $M$. Given any number $c>1$, we show that…

In this article, we consider currents given by the p-fold non-pluripolar product associated with a quasi-plurisubharmonic function of finite energy, and prove that normalized pull-backs of such currents converge to the Green (p, p)-current…

We solve the Dirichlet problem for the quaternionic Monge-Amp\`ere equation with a continuous boundary data and the right hand side in $L^p$ for $p>2$. This is the optimal bound on $p$. We prove also that the local integrability exponent of…

Let $\mathcal J$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=\log|f|+\mathcal O(1)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that…

We prove several approximation theorems of the complex Monge-Ampere operator on a compact Kahler manifold. As an application we give a new proof of a recent result of Guedj and Zeriahi on a complete description of the range of the complex…

Let $u$ be a plurisubharmonic function, defined on a neighbourhood of a point $x,$ such that the complex Monge-Amp\`ere operator is well-defined on $u.$ Suppose also that $u$ has a weak singularity, in the sense that the Lelong number of…

We introduce a synthetic approach to global pluripotential theory, covering in particular the case of a compact K\"ahler manifold and that of a projective Berkovich space over a non-Archimedean field. We define and study the space of…

We study the regularizing properties of complex Monge-Amp\`ere flows on a K\"ahler manifold $(X,\omega)$ when the initial data are $\omega$-psh functions with zero Lelong number at all points. We prove that the general Monge-Amp\`ere flow…

In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Amp\`ere equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal…

Let $(X,\omega)$ be a compact K\"ahler manifold. We prove that all Monge-Amp\`ere capacities are comparable. Using this we give an alternative direct proof of the integration by parts formula for non-pluripolar products recently proved by…

Our aim in this paper is to prove that if plurisubharmonic functions $u_1,. . . , u_n$, $v_1,. . ., v_n$ in the domain of definition of the complex Monge-Amp\`ere operator on a domain set $D\subset \mathbb{C}^n$ ($n\geq 1$) are such that…

On a smooth domain $\Omega\subset\subset\mathbb C^n$, we consider the Dirichlet problem for the complex Monge-Amp\`ere equation $((dd^cu)^n=fdV,\,u|_{b\Omega}\equiv\phi)$. We state the H\"older regularity of the solution $u$ when the…

We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Amp\`ere flow on a compact K\"ahler manifold, with the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere…

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Amp\`ere mass $\mathcal{E}(X, \theta, \phi)$, for a model potential $\phi$. We then investigate stability properties of this condition with respect to…

We characterize the class of probability measures on a compact Kahler manifold such that the associated Monge-Amp\`ere equation has a solution of finite pluricomplex energy. Our results are also valid in the big cohomology class setting.

In this paper, we explore some connections between Kobayashi geometry and the Dirichlet problem for the complex Monge--Amp\`ere equation. Among the results we obtain through these connections are: $(i)$~a theorem on the continuous extension…

In this note we study a complex Monge-Amp\`{e}re type equation of the form (dd^cu)^n = \frac {ke^{-u}dV}{\int e^{-u}dV}\,

We study the Parabolic complex Monge-Amp\'ere equation in a bounded strictly pseudoconvex domain in \mathbb{C}^n, with the boundary condition u=\varphi and the initial condition u=u_0. In this paper, we consider the case where \varphi is…

We make a systematic study of (quasi-)plurisubharmonic envelopes on compact K\"ahler 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…

We solve the Dirichlet problem for Monge-Amp\`ere equation for $(n-1)$-PSH functions possibly with degenerate right-hand side, through deriving a quantitative version of boundary estimate under the assumption of $(n-1)$-PSH subsolutions. In…