On Levi flat hypersurfaces with transversely affine foliation

A residue formula for meromorphic connections and applications to stable sets of foliations

“…where D = N ν=1 D ν is the irreducible decomposition. The rest of the proof is exactly as in [1,6]. We take an arbitrary smooth Hermitian metric h L of L, and Hermitian metrics h ν of the line bundle defined by D ν so that the curvature of h ν has support in X \ π −1 (M ).…”

“…Such Levi flat hypersurfaces contain either a compact leaf or are defined by a closed one-form. Using a residue formula that localizes the first Chern class to the singular locus of a logarithmic connection, obtained by [18], the authors in [1] prove the nonexistence of real analytic closed Levi flat hypersurfaces in compact Kähler surfaces whose Levi foliation is transversely affine and whose complement is 1-convex.…”

“…The residue formula stated above and Theorem 4.1 allow us to give a different proof of Brunella's conjecture in [2]. Also a residue formula recently given in [19] (see Theorem 3.1) simplifies the proof of the non-existence of real analytic Levi flat with transversely affine Levi foliation in [1]. We discuss these applications of residue formulae in Section 5.…”

Dynamical aspects of foliations with ample normal bundle

“…Once we can localize c 1 (N F ) to A, a contradiction easily follows from the ∂∂lemma and the maximum principle for strictly plurisubharmonic functions in the same way as in [2,8]…”

“…Using this connection ∇ hol , we would like to localize the first Chern class c 1 (N F ), not its square, to A. If ∇ hol was integrable, the desired localization would follow from the residue formula of integrable connections as in [2,8]. Due to the lack of a residue formula ready-to-use, we accomplish this localization via that of the first Atiyah class a 1 (N F ), inspired by [1].…”