Narrow Results

Let T be a positive closed current of bidegree (1,1) on a compact complex surface. We show that for all ε > 0, one can find a finite composition of blow-ups π such that π*T decomposes as the sum of a divisorial part and a positive closed current whose Lelong numbers are all less than ε.

We study the equation $\dot{u}=logdet(u_{\alpha \bar{\beta }})-Au+f(z,t)$ in $\mathrm{\Omega }\times (0,T)$, where $A\ge 0,T>0$ and $\mathrm{\Omega }$ is a bounded strictly pseudoconvex domain in ${ℂ}^n$, with the boundary condition $u=\phi$ and the initial condition $u=u_0$. In this paper, we consider the case, where $\phi$ is smooth and $u_0$ is an arbitrary plurisubharmonic function in a neighborhood of $\bar{\mathrm{\Omega }}$ satisfying $u_0{|}_{\partial \mathrm{\Omega }}=\phi (\cdot ,0)$.