Marco Abate, Universita' di Pisa

"Index Theorems for Holomorphic Maps and Foliations.
      Part I: A unified approach"

ABSTRACT:

In 1982, Camacho and Sad proved that if S is a compact leaf of a  
holomorphic foliation F on a complex surface M, then it is possible  
to associate to each singular point p of F in S a complex number (the  
index of F at p along S), depending only on the local behavior of F  
near p, so that the sum of the indeces is the first Chern class of
the normal bundle of S into M.

This theorem, which has profound implications in the theory of  
holomorphic foliations of complex surfaces, has later been  
generalized by Lehmann, Suwa and others to triples (M,F,S) where S is  
a (possibly singular) subvariety of a complex manifold M tangent to a  
(possibily singular) holomorphic foliation F of S; and, recently, by  
Camacho-Movasati-Sad and Camacho-Lehmann, to some cases where S  
is transversal to the foliation.

In 2004, Abate, Bracci and Tovena showed how to get a Camacho-Sad- 
like index theorem when the foliation F is replaced by a holomorphic  
self-map f of the complex manifold M leaving the subvariety S  
pointwise fixed. Furthermore, they obtained such a result both in the  
case when S is (in a suitable sense) tangent to the map f, and when S  
is (in a suitable sense) transversal to the map f, in the latter case  
under some interesting geometrical hypotheses on the embedding of S  
into M.

In this talk we shall describe the content of these index theorems,  
and describe a general argument reducing the proof of such an index  
theorem to the splitting of a suitable sequence of sheaves, thus  
providing a unified approach to all known (and some new) instances   
of Camacho-Sad-like index theorems.

Second Talk:

Francesca Tovena, Universita' di Roma Tor Vergata

"Index Theorems for Holomorphic Maps and Foliations.
      Part II: The case of holomorphic maps."

ABSTRACT:

In this talk we shall describe in detail the general argument  
presented in the previous talk in the particular case of holomorphic  
self-maps leaving a compact subvariety S of positive dimension  
pointwise fixed.

The main new tool is a section of a suitable vector bundle over S  
canonically associate to this situation, and having interesting  
dynamical interpretations.

Finally, we shall describe a couple of applications of these index  
theorems to the study of the dynamics of holomorphic self-maps.