58 research outputs found
A general approach to index theorems for holomorphic maps and foliations
No full textLet M be a smooth complex manifold, and S(subset of M) be a compact irreducible subvariety with dim(C) S > 0. Let be given either a holomorphic map f : M -> M with f(|S) = id(S), f not equal id(M), or a holomorphic foliation F on M: we describe an approach that can be applied to both map and foliation in order to obtain index theorems
Residual indices of holomorphic maps relative to singular curves of fixed points on surfaces
No full textLet M be a two-dimensional complex manifold and let be a holomorphic map that fixes pointwise a (possibly) singular compact reduced and globally irreducible curve . We give a notion of degeneracy of f at a point of C. It turns out that f is non-degenerate at one point if and only if it is non-degenerate at every point of C. When f is non-degenerate on C we define a residual index for f at each point of C. Then we prove that the sum of the indices is equal to the self-intersection number of C. © Springer-Verlag Berlin Heidelberg 2002
Poincaré-Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields
No full textWe 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 ∇ along F so that integral curves of Q are described by the geodesic flow of ∇ 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 neighborhood 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 neighborhood of the origin for a substantial class of 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
Index theorems for holomorphic self-maps
No full textLet be a complex manifold and a (possibly singular) subvariety of . Let be a holomorphic map such that restricted to is the identity. We show that one can associate to a holomorphic section of a sheaf related to the embedding of in and that such a section reads the dynamical behavior of along . In particular we prove that under generic hypotheses the canonical section induces a holomorphic action in the sense of Bott on the normal bundle of (the regular part of) in and this allows to obtain for holomorphic self-maps with non- isolated fixed points index theorems similar to Camacho-Sad, Baum-Bott and variation index theorems for holomorphic foliations. Finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self-maps of surfaces with a compact curve of fixed points
Poincaré-Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields
No full textWe first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincare-Bendixson theorem describing recurrence properties and omega-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 del(0) along F so that integral curves of Q are described by the geodesic flow of del(0) 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 neighborhood 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 neighborhood of the origin for a substantial class of 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)
Index theorems for pairs of holomorphic self-maps in the Lehmann-Suwa framework
No full textLet M be a n-dimensional complex manifold, let S be a globally irreducible compact analytic hypersurface with regular part S'=S-Sing(S), and let (f,g) be a pair of distinct holomorphic self-maps coinciding on S and such that g is a local biholomorphism over an open neighborhood of S'. We show that under certain hypotheses, on the pair (f,g) or on the way S' sits into M, we are able to define a 1-dimensional holomorphic foliation on S' and related partial holomorphic connections on some holomorphic vector bundles over S'. Consequently, we can obtain index theorems using the so-called Lehmann-Suwa machinery, which is based on localization of characteristic classes in Cech-de Rham cohomology
Formal normal forms for holomorphic maps tangent to the identity
No full textWe describe a procedure for constructing formal normal forms of holomorphic maps with a hypersurface of fixed points, and we apply it to obtain a complete list of formal normal forms for 2-dimensional holomorphic maps tangential to a curve of fixed points
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