Hypersurfaces quasi-invariant by codimension one foliations
- PDF / 414,852 Bytes
- 23 Pages / 439.37 x 666.142 pts Page_size
- 103 Downloads / 219 Views
Mathematische Annalen
Hypersurfaces quasi-invariant by codimension one foliations Jorge Vitório Pereira1 · Calum Spicer2 Received: 3 May 2018 / Revised: 1 March 2019 © The Author(s) 2019
Abstract We present a variant of the classical Darboux-Jouanolou Theorem. Our main result provides a characterization of foliations which are pull-backs of foliations on surfaces by rational maps. As an application, we provide a structure theorem for foliations on 3-folds admitting an infinite number of extremal rays. Mathematics Subject Classification 37F75 · 14E30
1 Introduction In this paper, we study codimension one singular holomorphic foliations on projective manifolds. More specifically, we introduce and investigate the concept of quasi-invariant hypersurfaces: an irreducible hypersurface H is quasi-invariant by a foliation F if it is not F invariant, but the restriction of the foliation F to H is an algebraically integrable foliation, i.e. every leaf of F|H is algebraic. 1.1 Statement of the main result Rational pull-backs of foliations on projective surfaces provide natural examples with infinitely many quasi-invariant divisors. Our main result shows that the existence of sufficiently many quasi-invariant hypersurfaces characterizes this class of foliations.
Communicated by Ngaiming Mok, Ph.D.
B
Calum Spicer [email protected] Jorge Vitório Pereira [email protected]
1
IMPA, Estrada Dona Castorina 110, Rio de Janeiro 22460-320, Brazil
2
Department of Mathematics, Imperial College London, London, England
123
J. V. Pereira, C. Spicer
Theorem A Let F be a codimension one holomorphic foliation on a projective manifold X . If F admits infinitely many quasi-invariant hypersurfaces then F is algebraically integrable, or F is a pull-back of a foliation on a projective surface under a dominant rational map. Indeed, we prove a stronger statement in Sect. 4, see Theorem 2. Theorem A has to be compared with, and was inspired by, Darboux-Jouanolou’s criterion for the algebraic integrability of codimension one foliations. We take the opportunity to revisit this criterion and we provide a small improvement of it in Sect. 2, see Theorem 1. 1.2 A version in positive characteristic In [15], an analogue of Darboux-Jouanolou’s Theorem was proved for codimension one foliations on smooth projective varieties defined over fields of positive characteristic, under the assumption that in the ambient variety every global differential 1-form is closed. This extra assumption was later proved to be superfluous by Brunella and Nicolau in [2]. We show that Brunella-Nicolau’s argument can be adapted to prove the following analogue of Theorem A. Theorem B Let X be a smooth projective variety defined over a field k of characteristic p > 0, and let F be a codimension one foliation on X . If F admits infinitely many quasi-invariant hypersurfaces then F is p-closed, or F is a pull-back of a foliation on a projective surface under a dominant rational map. The concepts used in the statement of Theorem B are recalled/presented in Sect. 5.1. 1.3 St
Data Loading...
