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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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