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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.

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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