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Projective varieties invariant by one-dimensional foliations. (English) Zbl 0983.32024

The author considers one-dimensional holomorphic foliations \({\mathcal F}^d\) of \(\mathbb{P}^n_\mathbb{C}\), \(n\geq 2\), given by a morphism \(\Phi:{\mathcal O}(1-d)\to T\mathbb{P}^n_\mathbb{C}\), \(d\geq 2\) such that the singular locus sing\(({\mathcal F}^d)\) of \({\mathcal F}^d\) has codimension \(\geq 2\). Besides he considers a smooth irreducible algebraic variety \(i:V^{n-k} \to\mathbb{P}^n_\mathbb{C}\) of codimension \(k\) which is invariant by \({\mathcal F}^d\) and such that \({\mathcal F}^d\) is non-degenerate along \(V\).
In this situation he proves a formula for the number of singularities of \({\mathcal F}^d\) in \(V^{n-k}\) which allows us to calculate this number by means of the Euler-Poincaré characteristic of algebraic varieties gotten from \(V^{n-k}\) by successively shaped generic hyperplane sections or by the classes \(\rho_i (V^{n-k})\) of \(V^{n-k}\). Moreover the number is positive.
If \(V^{n-k}\) is a complete intersection one gets a more special formula using Wronski functions. If in this situation \(n-k\) is odd one gets \[ d\geq {\rho_{n-k} (V^{n-k}) \over \rho_{n-k-1} (V^{n-k})}. \]

MSC:

32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32S65 Singularities of holomorphic vector fields and foliations
57R30 Foliations in differential topology; geometric theory
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