Sheaves associated to holomorphic first integrals

On the Poincaré problem for foliations of general type

“…To have some idea of these recent developments the interested reader should consult the papers by Carnicer [3], Soares [11,12], Brunella-Mendes [2], Esteves [5] and Zamora [13,14].…”

“…Since 1991 paper of Cerveau and Lins Neto [4], this problem has received the attention of many mathematicians. The interested reader should consult the papers by Carnicer [3], Soares [12,13], Brunella-Mendes [2], Esteves [5] and Zamora [14,15] to have some idea of these recent developments.…”

Foliations with isolated singularities on Hirzebruch surfaces

“…Considering holomorphic foliations by curves with singularities (foliations in the sequel) on the complex projective plane have produced important advances in the knowledge of those systems [5, 9, 11, 15-20, 29, 32]. Foliations can be defined on another varieties extending the problems from the projective plane to those varieties [10,12,21,26,38,39]. Focusing on foliations on surfaces, Hirzebruch surfaces S δ with δ ̸ = 1 (see Section 2 for our notation) constitute jointly with the projective plane the classical minimal rational surfaces, and the study of foliations on them is the first single step after that on the projective plane.…”

Analytic varieties invariant by foliations and Pfaff systems

“…Mendes [17], E. Esteves and S. Kleiman [55], V. Cavalier and D. Lehmann [28]. This problem also has been considered in other varieties such as: del Pezzo surfaces and K3 surfaces [61], weighted projective spaces [12,45], multiprojective spaces [44], toric varieties [92], projective manifolds with Picard number equal to one [17], surfaces with trivial Picard group [3], varieties over an algebraically closed field of arbitrary characteristic [55]. It follows from a celebrated theorem by Jouanolou [69] that foliations on projective plane, of degree at least 2, with some invariant algebraic curve are rare.…”