1-convex manifolds are \(p\)-Kähler
A complex manifold \(X\) is called \(p\)-Kähler if it carries a closed transverse (i.e. strictly weakly positive) \((p,p)\)-form \(\omega\). \(\omega\) is called a \(p\)-Kähler form. Kähler manifolds are 1-Kähler and also \(p\)-Kähler for each \(p\) such that \(1 \leq p \leq\dim X\), therefore this notion is relevant in the non-Kähler case. The authors prove that if \(X\) is a 1-convex manifold possessing an exceptional set \(S\) with \(\dim S=k\), then \(X\) is \(p\)-Kähler for every \(p\) greater than \(k\). Also, \(X\) is \(k\)-Kähler iff the ``fundamental class'' of \(S\) does not vanish. Several further interesting discussions and examples are also presented.
scientific article; zbMATH DE number 1868559
scientific article