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Đề tài " Index theorems for holomorphic self-maps "

The usual index theorems for holomorphic self-maps, like for instance the classical holomorphic Lefschetz theorem (see, e.g., [GH]), assume that the fixed-points set contains only isolated points. The aim of this paper, on the contrary, is to prove index theorems for holomorphic self-maps having a positive dimensional fixed-points set. The origin of our interest in this problem lies in holomorphic dynamics. A main tool for the complete generalization to two complex variables of the classical Leau-Fatou flower theorem for maps tangent to the identity achieved in [A2] was an index theorem for holomorphic self-maps of a complex surface fixing. | Annals of Mathematics Index theorems for holomorphic self-maps By Marco Abate Filippo Bracci and Francesca Tovena Annals of Mathematics 159 2004 819 864 Index theorems for holomorphic self-maps By Marco Abate Filippo Bracci and Francesca Tovena Introduction The usual index theorems for holomorphic self-maps like for instance the classical holomorphic Lefschetz theorem see e.g. GH assume that the fixed-points set contains only isolated points. The aim of this paper on the contrary is to prove index theorems for holomorphic self-maps having a positive dimensional fixed-points set. The origin of our interest in this problem lies in holomorphic dynamics. A main tool for the complete generalization to two complex variables of the classical Leau-Fatou flower theorem for maps tangent to the identity achieved in A2 was an index theorem for holomorphic self-maps of a complex surface fixing pointwise a smooth complex curve S. This theorem later generalized in BT to the case of a singular S presented uncanny similarities with the Camacho-Sad index theorem for invariant leaves of a holomorphic foliation on a complex surface see CS . So we started to investigate the reasons for these similarities and this paper contains what we have found. The main idea is that the simple fact of being pointwise fixed by a holomorphic self-map f induces a lot of structure on a possibly singular subvariety S of a complex manifold M. First of all we shall introduce in 3 a canonically defined holomorphic section Xf of the bundle TMIs Ng Vf where N s is the normal bundle of S in M here we are assuming S smooth however we can also define Xf as a section of a suitable sheaf even when S is singular see Remark 3.3 but it turns out that only the behavior on the regular part of S is relevant for our index theorems and Vf is a positive integer the order of contact of f with S measuring how close f is to being the identity in a neighborhood S see 1 . Roughly speaking the section Xf describes the directions