In this paper we introduce, via a Phragmen-Lindelof type theorem, a maximal plurisubharmonic function in a strongly pseudoconvex domain. We call such a function the {sl pluricomplex Poisson kernel} because it shares many properties with the classical
Poisson kernel of the unit disc. In particular, we show that such a function is continuous, it is zero on the boundary except at one boundary point where it has a non-tangential simple pole, and reproduces pluriharmonic functions. We also use such a function to obtain a new intrinsic version of the classical Julias Lemma and Julia-Wolff-Caratheodory Theorem.
Let $(X, omega)$ be a compact Kahler manifold of complex dimension n and $theta$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class ${ theta }in H^{1,1}(X, mathbb{R})$ is pseudoeffective. Let $varphi$ be a $theta$-psh function
, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $omega$ such that $varphi leq f. $ Then the non-pluripolar measure $theta_varphi^n:= (theta + dd^c varphi)^n$ satisfies the equality: $$ {bf{1}}_{{ varphi = f }} theta_varphi^n = {bf{1}}_{{ varphi = f }} theta_f^n,$$ where, for a subset $Tsubseteq X$, ${bf{1}}_T$ is the characteristic function. In particular we prove that [ theta_{P_{theta}(f)}^n= { bf {1}}_{{P_{theta}(f) = f}} theta_f^nqquad {rm and }qquad theta_{P_theta[varphi](f)}^n = { bf {1}}_{{P_theta[varphi](f) = f }} theta_f^n. ]
In recent papers Wu-Yau, Tosatti-Yang and Diverio-Trapani, used some natural differential inequalities for compact Kahler manifolds with quasi negative holomorphic sectional curvature to derive positivity of the canonical bundle. In this note we study the equality case of these inequalities.
We show that if a compact complex manifold admits a Kahler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.
Let $alpha$ be a big class on a compact Kahler manifold. We prove that a decomposition $alpha=alpha_1+alpha_2$ into the sum of a modified nef class $alpha_1$ and a pseudoeffective class $alpha_2$ is the divisorial Zariski decomposition of $alpha$ if
and only if $operatorname{vol}(alpha)=operatorname{vol}(alpha_1)$. We deduce from this result some properties of full mass currents.
Let $T$ be a random field invariant under the action of a compact group $G$. In the line of previous work we investigate properties of the Fourier coefficients as orthogonality and Gaussianity. In particular we give conditions ensuring that independe
nce of the random Fourier coefficients implies Gaussianity. As a consequence, in general, it is not possible to simulate a non-Gaussian invariant random field through its Fourier expansion using independent coefficients.
We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.
We present a classification of 2-dimensional, taut, Stein manifolds with a proper $R$-action. For such manifolds the globalization with respect to the induced local $C$-action turns out to be Stein. As an application we determine all 2-dimensional ta
ut, non-complete, Hartogs domains over a Riemann surface.
In this note we use the divisorial Zariski decomposition to give a more intrinsic version of the algebraic Morse inequalities.
We show that for every smooth generic projective hypersurface $Xsubsetmathbb P^{n+1}$, there exists a proper subvariety $Ysubsetneq X$ such that $operatorname{codim}_X Yge 2$ and for every non constant holomorphic entire map $fcolonmathbb Cto X$ one
has $f(mathbb C)subset Y$, provided $deg Xge 2^{n^5}$. In particular, we obtain an effective confirmation of the Kobayashi conjecture for threefolds in $mathbb P^4$.
