Related papers: Polynomial hulls and positive currents

In this note we study the existence of the Lelong-Demailly number of a negative plurisubharmonic current with respect to a positive plurisubharmonic function on an open subset of $\C^n$. Then we establish some estimates of the…

Dinh--Sibony theory of tangent and density currents is a recent but powerful tool to study positive closed currents. Over twenty years ago, Alessandrini and Bassanelli initiated the theory of the Lelong number of a positive plurisubharmonic…

Recently the authors showed that there is a robust potential theory attached to any calibrated manifold (X,\phi). In particular, on X there exist \phi-plurisubharmonic functions, \phi-convex domains, \phi-convex boundaries, etc., all…

In this paper we study the existence of the directional Lelong-Demailly numbers of positive plurisubharmonic or plurisuperharmonic currents. We prove the independence of these numbers to the system of coordinates. Moreover these numbers…

In this paper, we study the existence of the tangent cone to a positive plurisubharmonic or plurisuperharmonic current with a suitable condition. Some Estimates of the growth of the Lelong functions associated to the current and to its…

If $(X,J)$ is an almost complex manifold, then a function $u$ is said to be plurisubharmonic on $X$ if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is…

In this paper we study the existence of Lelong numbers of $m-$subharmonic currents of bidimension $(p,p)$ on an open subset of $\Bbb C^n$, when $m+p\geq n$. In the special case of $m-$subharmonic function $\varphi$, we give a relationship…

In this article, we study the order of a positive pluriharmonic current and we compare it with either the order of the concurrent slices or the directionnel orders of the current. Therefore some estimates of the growth of the…

Each extreme edge of the Newton diagram of a plurisubharmonic polynomial on $\mathbb{C}^2$ gives rise to a plurisubharmonic polynomial. It is tempting to believe that the union of the extreme edges or the convex hull of said union will do…

This is a survey of results, both classical and recent, on behaviour of plurisubharmonic functions near their $-\infty$-points, together with the related topics for positive closed currents.

Let A be a closed polar subset of a domain D in the complex plane C. We give a complete description of the pluripolar hull in D X C of the graph of a holomorphic function defined on D A. To achieve this, we prove for pluriharmonic measure…

Let $T$ be a positive closed current of bidimension $(p,p)$ with unit mass on the complex projective space $\mathbb P^n$. For certain values of $\alpha$ and $\beta = \beta(p, \alpha)$ we show that if $T$ has enough points where the Lelong…

Let $X$ be a complex manifold of dimension $k,$ and $(V,\omega)$ be a K\"ahler submanifold of dimension $l$ in $X,$ and $B\Subset V$ be a domain with $\mathcal{C}^2$-smooth boundary. Let $T$ be a positive plurisubharmonic current on $X$…

Let $X$ be a compact K\"ahler manifold. We study plurisupported currents on $X$, i.e. closed, positive $(1,1)$-currents which are supported on a pluripolar set. In particular, we are able present a technical generalization of…

Let $T$ be a positive closed current of bidegree $(1,1)$ on a multiprojective space $X={\mathbb P}^{n_1}\times\ldots\times{\mathbb P}^{n_k}$. For certain values of $\alpha$, which depend on the cohomology class of $T$, we show that the set…

In this paper, we first establish a Poincar\'e-Lelong type formula in the almost complex setting. Then, after introducing the notion of $J$-analytic subsets, we study the restriction of a closed positive current defined in an almost complex…

We extend certain classical theorems in pluripotential theory to a class of functions defined on the support of a $(1,1)$-closed positive current $T$, analogous to plurisubharmonic functions, called $T$-plurisubharmonic functions. These…

Let $T$ be a positive closed current of unit mass on the complex projective space $\mathbb P^n$. For certain values $\alpha<1$, we prove geometric properties of the set of points in $\mathbb P^n$ where the Lelong number of $T$ exceeds…

Demailly showed that the Hodge conjecture is equivalent to the statement that any (p,p)-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents associated to subvarieties,…

Let $f$ be a polynomial automorphism of ${\Bbb C}^k$ of degree $\lambda$, whose rational extension to ${\Bbb P}^k$ maps the hyperplane at infinity to a single point. Given any positive closed current $S$ on ${\Bbb P}^k$ of bidegree (1,1),…