We prove several index theorems for holomorphic self-maps having positive-dimensional fixed points set. To do so we show that the fixed points set of a holomorphic self-map has a surprisingly rich structure, expressed by canonically defined meromorphic connections and bundle maps. Finally, we present some applications to holomorphic dynamics

We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincaré-Bendixson theorem describing recurrence properties and ω-limit sets of geodesics for a meromorphic connection on P 1 (C). We then show how to associate to a homogeneous vector field Q in C n a rank 1 singular holomorphic foliation F of P n−1 (C) and a (partial) meromorphic connection ∇ o along F so that integral curves of Q are described by the geodesic flow of ∇ o along the leaves of F , which are Riemann surfaces. The combination of these results yields powerful tools for a detailed study of the dynamics of homogeneous vector fields. For instance, in dimension two we obtain a description of recurrence properties of integral curves of Q, and of the behavior of the geodesic flow in a neighbourhood of a singularity, classifying the possible singularities both from a formal point of view and (for generic singularities) from a holomorphic point of view. We also get examples of unexpected new phenomena, we put in a coherent context scattered results previously known, and we obtain (as far as we know for the first time) a complete description of the dynamics in a full neighbourhood of the origin for a substantial class of 2dimensional holomorphic maps tangent to the identity. Finally, as an example of application of our methods we study in detail the dynamics of quadratic homogeneous vector fields in C 2 .

We describe a general construction providing index theorems localizing the Chern classes of the normal bundle of a subvariety inside a complex manifold. As particular instances of our construction we recover both Lehmann-Suwa's generalization of the classical Camacho-Sad index theorem for holomorphic foliations and our index theorem for holomorphic maps with positive dimensional fixed point set. Furthermore, we also obtain generalizations of recent index theorems of Camacho-Movasati-Sad and Camacho-Lehmann for holomorphic foliations transversal to a subvariety.

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

This paper studies the embeddings of a complex submanifold S inside a complex manifold M; in particular, we are interested in comparing the embedding of S in M with the embedding of S as the zero section in the total space of the normal bundle N(S) of S in M. We explicitly describe some cohomological classes allowing to measure the difference between the two embeddings, in the spirit of the work by Grauert, Griffiths, and Camacho, Movasati and Sad; we are also able to explain the geometrical meaning of the separate vanishing of these classes. Our results hold for any codimension, but even for curves in a surface we generalize previous results due to Laufert and Camacho, Movasati and Sad. (C) 2008 Elsevier Inc. All rights reserved