On the Poincaré inequality for one-dimensional foliations

Abstract

Let d be the degree of an algebraic one-dimensional foliation $\mathcal F$ on the complex projective space ${\mathbb P}_n$ (i.e. the degree of the variety of tangencies of the foliation with a generic hyperplane). 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 ${\mathcal B}(\Gamma)=1$ (instead of 0) 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.