Poincare problem for divisors invariant by one-dimensional foliations on smooth algebraic variety

TL;DR

This paper investigates bounds on the degree of divisors invariant under one-dimensional holomorphic foliations on smooth algebraic varieties, establishing conditions for the existence of rational first integrals based on invariants and algebraic solutions.

Contribution

It provides new bounds and criteria for the existence of rational first integrals for foliations on algebraic varieties, extending previous results in the Poincaré problem context.

Findings

Bound on the degree of invariant divisors in terms of foliation degree and invariants.

Existence of a degree bound for rational first integrals when the number of invariant curves exceeds a threshold.

Criteria linking algebraic solutions' degree and genus to the existence of rational first integrals.

Abstract

In this paper we consider the question of bounding the degree of an divisor $D$ invariant by a $\F$ holomorphic foliation, without rational first integral, on smooth algebraic variety $X$ in terms of degree of $\F$ and some invariants of $D$ and $X$. Particularly, if $\F$ is a foliation of degree $d$ on $\mathbb{P}_{\mathbb{C}}^2$, whose the number of invariants curves is greater that ${k+2\choose k}$, we show that there exist a number $\mathcal{M}(d,k)$ such that if $k>\mathcal{M}(d,k),$ then $\F$ admits a rational first integral of degree $\leq k$. Moreover, there exist a number $\mathscr{G}(d,k)$, such that if $\F$ has an algebraic solution of degree $k$ and genus smaller than $\mathscr{G}(d,k)$, then it has a rational first integral of degree $\leq k$.