On some foliations on ruled surfaces
Let \(X\) be a two-dimensional complex manifold, \(\theta_ X\) the sheaf of germs of holomorphic functions on \(X\) and \(\Theta_ X\) the sheaf of germs of holomorphic vector fields on \(X\). \(C\) a closed Riemann surface, \(\mathcal F\) a one-dimensional foliation on \(X\) (an invertible subsheaf of \(\Theta_ X\) with singular locus of codimension strictly greater than one), and \(N\) a submanifold of \(X\), invariant to respect to \(\mathcal F\) (then defined by a coherent ideal sheaf \({\mathcal J} \subset \Theta_ X\) such that every \(p \in N\), \({\mathcal F}_ p {\mathcal J}_ p \subset {\mathcal J}_ p\)). Let \(X @>\pi>> C\) be a ruled surface, then \(\pi\) a \(\mathbb{P}^ 1\)-bundle and let be \(f\) their fibre. Denote \(C_ 0\) a section from \(\pi\). Let be \(C^ 2_ 0 = \) the minimum of self-intersection numbers of sections of \(\pi\) and take \(e = -C_ 0^ 2\). If \(\text{Num }X\) denotes the group of numerically equivalent classes of divisors on \(X\), from the commutative diagram of the cohomology long exact sequences associate to the exponential sequences on \(X\) and \(C\), follow the isomorphism \(\text{Num }X = H^ 2(X, \mathbb{Z}) = \mathbb{Z} c (C_ 0) \oplus \mathbb{Z} c(f) = \mathbb{Z} \oplus \mathbb{Z} = \mathbb{Z}^ 2\), where \(c(C_ 0)\) and \(c(f)\) are the images by the Chern map of the holomorphic line bundle defined by the divisors \(C_ 0\) and \(f\). Let \(C_ 1 =_{\text{num}} aC_ 0 + bf\) be an irreducible curve on \(X\), with ``\(=_{\text{num}}\)'' the numerically equivalence of divisors on \(X\). The author states the following Main Theorem. Let \(X @>\pi>> C\), \(C_ 0\) and \(f\) as above. Assume that a foliation \(\mathcal F \subset \Theta_ X\) on \(X\) leaves an irreducible curve \(C_ 1 =_{\text{num}} aC_ 0 + bf\) with \(a > 0\) on \(X\) invariant and has no singularities on \(C_ 1\). Then one of the following is the case. I) \(e = 0\) and \(b = 0\). II) \(e < 0\), \(a = 2\) and \(b = 1/2e\) \(a \in \mathbb{Z}\). The demonstration follows from the Camacho-Sad index formula and the general theory of ruled surfaces. The details will be published elsewhere. The author gives two examples.
scientific article; zbMATH DE number 597322
scientific article