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On the Poincaré inequality for one-dimensional foliations. (English) Zbl 1097.32010
Authors’ abstract: Let \(d\) be the degree of an algebraic one-dimensional foliation \(\mathcal{F}\) on the complex projective space \(\mathbb{P}_n\). Let \(\Gamma\) be an algebraic solution of degree \(\delta\), and geometrical genus \(g\).
We prove, in particular, the inequality \(( d-1)\delta +2-2g\geq\mathcal{B}(\Gamma)\), where \(\mathcal{B}(\Gamma)\) denotes the total number of locally irreducible branches through singular points of \(\Gamma\) when \(\Gamma\) has singularities, and \(\mathcal{B}(\Gamma) =1\) when \(\Gamma\) is smooth.
Equivalently, when \(\Gamma =\bigcap_{\lambda =1}^{n-1}S_{\lambda}\) is the complete intersection of \(n-1\) algebraic hypersurfaces \(S_{\lambda}\), we get \(( d+n-\sum_{\lambda =1}^{n-1}\delta _{\lambda})\delta\geq\mathcal{B}(\Gamma)-\mathcal{E}(\Gamma)\), where \(\delta _{\lambda}\) denotes the degree of \(S_{\lambda}\) and \(\mathcal{E}(\Gamma) =2-2g+(\sum _{\lambda}\delta_{\lambda} -( n+1))\delta\) the correction term in the genus formula. These results are also refined when \(\Gamma\) is reducible.

MSC:
32S65 Singularities of holomorphic vector fields and foliations
14M10 Complete intersections
19E20 Relations of \(K\)-theory with cohomology theories
32S20 Global theory of complex singularities; cohomological properties
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