Nonexistence of certain Levi flat hypersurfaces in Kähler manifolds from the viewpoint of positive normal bundles (Q355141): Difference between revisions

In this paper, which is dedicated to the memory of Marco Brunella, the author combines the \(L^2\) technique for solving the \(\overline\partial\)-equation with a method by Brunella for proving two nonexistence theorems for Levi flat hypersurfaces.NEWLINENEWLINEIn both theorems, \(X\) is a compact Kähler manifold of dimension at least \(3\) and \(M\) is a Levi-flat hypersurface in \(X\). Then certain positiveness conditions of the curvature are proved to be impossible. Namely:NEWLINENEWLINE1. Suppose \(M\) is of class \(\mathcal C^{2,\alpha}\), \(\alpha>0\), and \(M\) is the union of leaves of a holomorphic foliation \(\mathcal F\) in a neighbourhood \(U\) of \(M\). Then the normal bundle of \(\mathcal F\) does not admit any fiber metric whose curvature form restricted to the tangent space of \(\mathcal F\) is semipositive of rank at least \(2\) at every point of \(M\).NEWLINENEWLINE2. Suppose \(M\) is of class \(\mathcal C^\infty\). Then the holomorphic normal bundle of \(M\) does not admit a fiber metric whose curvature form is semipositive of rank at least \(2\) on the holomorphic tangent space of \(M\).