Monge-Ampère currents over pseudoconcave spaces (Q5947130)

The author studies the growth of Monge-Ampère masses along pseudoconcave ends in a complex manifold. In particular, as applications, he proves the following two Hartogs-type extension theorems. Let \(V\) be a projective manifold with \(n:=\dim V\geq 2\). (1) Let \(H\) be a complex hypersurface in \(V\) such that \(V\setminus H\) is pseudoconcave and let \(X, M\subset V\) be open neighborhoods of \(H\) with \(X\subset\subset M\). Let \(\omega\) be a closed positive \((1,1)\)-current on \(M\setminus H\) which has local locally bounded potentials. Then \(\int_{\overline X\setminus H}\omega^n<+\infty\) and \(\omega^k\) extends through \(H\) as a closed positive current, \(k=1,\dots,n\). (2) Let \(U\subset V\) be an open set such that \(U=\text{int} \overline U\) and \(V\setminus\overline U\) is a pseudoconcave domain. Let \(W\) be a neighborhood of \(\partial U\). Then any meromorphic mapping \(W\rightarrow\mathbb P^N\) extends meromorphically to \(U\).