GEOMETRY AND TOPOLOGY OF CAUSTICS — CAUSTICS ’06 BANACH CENTER PUBLICATIONS, VOLUME 82 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2008 GLOBAL STRUCTURE OF HOLOMORPHIC WEBS ON SURFACES VINCENT CAVALIER and DANIEL LEHMANN D´epartement des Sciences Math´ematiques, CP 051, Universit´e de Montpellier II Place Eug`ene Bataillon, F-34095 Montpellier Cedex 5, France E-mail: cavalier@math.univ-montp2.fr, lehmann@math.univ-montp2.fr Abstract. The webs have been studied mainly locally, near regular points (see a short list of references on the topic in the bibliography). Let d be an integer ≥ 1. A d-web on an open set U of C 2 is a differential equation F (x, y, y ′ ) = 0 with F (x, y, y ′ )= P d i=0 a i (x, y)(y ′ ) d−i , where the coefficients a i are holomorphic functions, a 0 being not identically zero. A regular point is a point (x, y) where the d roots in y ′ are distinct (near such a point, we have locally d foliations mutually transverse to each other, and caustics appear through the points which are not regular). It happens that many concepts on local webs may be globalized, but not always in an obvious way, and under the condition that they do not depend on local coordinates. The aim of this paper is to make these facts precise and to define the tools necessary for a global study of webs on a holomorphic surface, and in particular on the complex projective plane P 2 . Moreover new concepts, inducing new problems, will appear, such as the dicriticality, the irreducibility or the quasi-smoothness, which have no interest locally near a regular point of the web. 1. Global definition of a web. First of all, we homogenize the equation in the abstract, for allowing the contact elements to be “vertical” (this notion does not make sense by change of local coordinates), and write the differential equation  = 0, where  = d  i=0 a i (x, y)(dx) i (dy) d−i is now a homogeneous polynomial of degree d on U (removing also the condition a 0 ≡ 0). Moreover, if we multiply  by a holomorphic non-vanishing function, we do not change the solutions of the differential equation. Hence, gluing together local webs defined as above, we get the following global definition. 2000 Mathematics Subject Classification : 14C21, 53A60. Key words and phrases : global web, dicriticality, type, degree, indistinguishability, quasi- smoothness, Blaschke curvature, Chern curvature, abelian relation. This paper is a summary of [CaLe], without proofs. [35] c  Instytut Matematyczny PAN, 2008