On the neighborhood of a compact complex curve with topologically trivial normal bundle

Neighborhood of a Rational Curve with a Node

“…Let d(p) denote the distance of p e S from the curve C with respect to some Riemannian metric These results correspond to Theorems 1 and 2 in [8] (for smooth curves of finite type). We note that, in the present case, the neighborhood of C admits plurisubharmonic functions with slower growth than in the case of [8], Theorems 1 and 2 are proved in § 3, after some preliminaries in § 1 and § 2. We consider, in § 4, such curves in compact complex surfaces.…”

“…When (C 2 )=0, this topological condition alone is insufficient to derive analytic conclusions. For a smooth curve C with (C 2 ) = 0, we obtained some conditions for the existence of a fundamental system of strongly pseudoconcave or pseudoflat neighborhoods in [8] (see also Neeman [5]). …”

Weakly complete compex surfaces

“…Another class of example is provided by total spaces of some complex line bundles over compact Riemann surfaces (see also [26]). …”

“…Moreover, the only result which holds in arbitrary dimension heavily employs the hypothesis of homogeneity with respect to automorphisms. It is also worth noticing that other results related to this problem, like the classification of holomorphic foliations or Ueda's results ( [26]), are fully understood and developed only in dimension 2.…”

A preparation theorem for codimension-one foliations

“…In dimension 2, when F is defined by a vector field X, it is still possible to extend X on a 2-dimensional tubular neighborhood M of an embedded sphere C but it is not possible to construct the C-fibration at the same time. Here, we need the Rigidity Theorem of V. I. Savelev [17] (see also [21] …”