Compositio Math. 142 (2006) 529–540 doi:10.1112/S0010437X05001764 On the Poincar´e inequality for one-dimensional foliations Vincent Cavalier and Daniel Lehmann Abstract Let d be the degree of an algebraic one-dimensional foliation F on the complex projective space P n (i.e. the degree of the variety of tangencies of the foliation with a generic hyper- plane). Let Γ be an algebraic solution of degree δ, and geometrical genus g. We prove, in particular, the inequality (d − 1)δ +2 − 2g  B(Γ), where B(Γ) denotes the total number of locally irreducible branches through singular points of Γ when Γ has singularities, and B(Γ) = 1 (instead of 0) when Γ is smooth. Equivalently, when Γ =  n−1 λ=1 S λ is the complete intersection of n − 1 algebraic hypersurfaces S λ , we get (d + n − ∑ n−1 λ=1 δ λ )δ  B(Γ) −E (Γ), where δ λ denotes the degree of S λ and E (Γ) = 2 − 2g +( ∑ λ δ λ − (n + 1))δ the correction term in the genus formula. These results are also refined when Γ is reducible. 1. Introduction In connection with the existence of first integrals, Poincar´e raised the question of bounding the degree δ of an algebraic solution Γ for an algebraic differential system F on the complex projective plane P 2 , in terms of the degree d of F . This is not possible without further conditions on F or on Γ. For example, Lins Neto proved in [Lin02] that the problem has no solution in the presence of dicritical singularities, i.e. of singularities through which there are infinitely many germs of separatrices (see Example 5.6 below). In fact, the inequality d +2 − δ  0 has been proved by Cerveau and Lins Neto [CL91] (see also [Soa01]) when Γ has only nodal singularities, and by Carnicer [Car94] when the foliation has no dicritical singularity. Moreover, Brunella [Bru97] recovered Carnicer’s result by observing that the negativity of the GSV-indices (see [GSV91]) is an obstruction to the above inequality, and proving that these indices are always non-negative in the non-dicritical case. Carnicer and Campillo [CC97] proved also that there exists some non-negative integer a, depending on conditions imposed on F or on Γ, such that d +2 − δ  −a. In higher dimension (i.e. for one-dimensional algebraic foliations on the complex projective space P n leaving invariant an algebraic curve Γ), the inequality (d+n− ∑ n−1 λ=1 δ λ )  1 has been proved by Soares [Soa00], when Γ is the complete intersection  n−1 λ=1 S λ of n − 1 algebraic hypersurfaces S λ of degree δ λ , under the further conditions that Γ be smooth, and the restriction of the foliation to Γ has non-degenerate singularities. More generally, in [Soa97, Soa00] he gave a lower bound for the degree of the algebraic foliations leaving invariant a smooth submanifold of P n , under conditions of non-degeneracy of the foliation. Also, Esteves and Kleiman [EK03] proved the inequality (d − 1)(δ − 1) − 2g  1 − r(Γ), r(Γ) denoting the number of globally irreducible components. In this paper, we consider the case of curves with any kind of singularity, in any dimension. Let F be the one-dimensional holomorphic foliation on a holomorphic manifold M defined by a Received 22 November 2004, accepted in final form 18 May 2005. 2000 Mathematics Subject Classification 57R20, 57R25, 19E20. Keywords: degree of an algebraic foliation, Poincar´e inequality, genus formula, GSV-index, residues. This journal is c  Foundation Compositio Mathematica 2006. https://doi.org/10.1112/S0010437X05001764 Published online by Cambridge University Press