Wedge product of positive currents and balanced manifolds
The authors study some wedge products of currents on Kählerian and balanced manifolds. Theorem 2.2. Let \(Y\) be a proper analytic subset of a complex manifold \(X\). Let \(S\) be a positive closed \((1,1)\)-current on \(X\), smooth on \(X-Y\), and let \(T\) be a positive pluriharmonic \((k,k)\)-current on \(X\), with \(k+ \dim Y< \dim X.\) Then there exists a unique current on \(X\), denoted by \(S\wedge T\), with the following property: If \(g\) is a solution if \(S=i\partial\overline{\partial}g\) in an open subset \(U\subset X\), and if \(\{g_j\}\) is a sequence of smooth purisubharmonic metrics on \(U\), which converge to \(g\) in \(C^\infty(U-Y)\), then \(\lim_j\partial\overline{\partial}g_j\wedge T=S\wedge T\) in \(U\). Theorem 3.4. If \(M\) is a compact complex manifold of dimension \(n\geq 3\), and \(C\) is an irreducible curve in \(M\) such that \(M-C\) is Kähler, then either \(M\) is Kähler itself or \([C]\) is the component of a boundary and \(M\) is \(p\)-Kähler for every \(p\geq 2\), or \([C]\) is part of the component of a boundary and \(M\) belongs to the Fujiki class \(\mathcal{C}\), i.e., \(M\) is bimeromorphic to a Kähler manifold. In all cases, \(M\) is balanced.
scientific article; zbMATH DE number 5281226