Levi problem in complex manifolds
Let \((M,\omega)\) be a complex Hermitian manifold and let \(U=\{z\in U_1: r(z)<0\}\subset\subset M\) be a \(\mathcal C^\infty\)-smooth pseudoconvex domain, where \(U_1\) is a neighborhood of \(\overline U\) and \(dr\neq0\) on \(\partial U\). A positive current \(T\) of bidimension \((1,1)\), supported by \(\partial U\), is said to be a \textit{Levi current} if \(T\wedge\partial r=0\), \(T\wedge\partial\overline\partial r=0\), \(i\partial\overline\partial T=0\), \(\langle T,\omega\rangle=1\). The main results of the paper are the following three theorems related to the Levi problem: -- If \(\partial U\) has no Levi current, then \(U\) is a proper modification of a Stein manifold. Moreover, there is a smooth function \(v\) such that if \(\varrho:=re^{-v}\), then there exists an \(\eta>0\) such that for \(\widehat\varrho:=-(-\varrho)^\eta\) we have \(i\partial\overline\partial\widehat\varrho\geq c|\widehat\varrho|\omega\) on \(U\setminus K\), where \(K\) is compact. -- Assume that there exist a compact \(K\subset U\) and a bounded function \(v\) on \(U\setminus K\) such that \(i\partial\overline\partial v\geq\omega\) on \(U\setminus K\). Then \(U\) is a proper modification of a Stein manifold and admits a bounded exhaustion function which is strictly plurisubharmonic outside of a compact set. -- Assume that \(U\) admits a continuous exhaustion function \(\varphi\), \(U\) is not a proper modification of a Stein manifold, and \(U\) contains at most finitely many compact varieties of positive dimension. Then there exists \(t_0\in\mathbb R\) such that for every \(t>t_0\) the level set \(\{\varphi=t\}\) has a Levi current \(T_t\). The author gives also a foliated version of the above results.
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