Índices of holomorphic fields and applications. 23rd Brazilian mathematics colloquium, Rio de Janeiro, Brazil, July 22--27, 2001. (Q2754863)

The main topic of this publication is concerned with one complex dimensional foliations on complex manifolds. Written in Portuguese, which can also be read by Spanish speakers, and directed to graduate students and non-specialist in the area, this book is divided into two parts (divided each one into chapters). The first part gives basic concepts and definitions (complex manifolds, vector fiber bundles, Chern's class, etc.) and the second one introduces holomorphic foliations (especially for complex surfaces and projective spaces). The authors claim in the introduction that this is a book for students at the beginning of a Ph.D. degree or thinking in going into that program. For the first chapter, it is recommended to have a preparation in several complex variables and at least a course in real differentiable manifolds. Also, the knowledge of simplicial homology groups and De Rham cohomology are in some way needed. A good thing is that the authors make a lot of examples and calculations to make results clear. NEWLINENEWLINENEWLINEThe second chapter involves vector fiber bundles and many examples which do not permit the reader to go into darkness. Two fiber vector bundles \(\pi_{E}:E \to X\) and \(\pi_{F}:F \to X\) are said to be isomorphic if there is a homeomorphism \(\varphi:E \to F\) preserving fibers, \(\pi_{F} \circ \varphi = Id_{X} \circ \pi_{E}\) and \(\varphi\) restricted to the fibers is an isomorphism of vector spaces. Later, in chapter 4 the space of isomorphism classes of line holomorphic bundles, denoted by \(Pic(X)\) and called the Picard group of \(X\), is defined. In chapter 3, connections of fiber bundles are discussed and the Chern's class is introduced. Rank one holomorphic fiber bundles are considered. In chapter 4, they introduce the concept of a sheaf of abelian groups on topological spaces. Then the cohomology of sheaves is defined and many of the properties deduced. The main result is Dolbeault's theorem which says that for a complex manifold \(M\) the Dolbeaut cohomology group of order \((p,q)\) is isomorphic to the \(q\)-th cohomology group of the sheaf of holomorphic \(p\)-forms. Then it is observed that we can identify the Picard group of \(M\) with the first cohomology group of \(M\) with respect to the sheaf of non-vanishing holomorphic (germs of) functions. In chapter 5, the last one of the first part, divisors are introduced (linear combinations of irreducible one complex dimensional analytic sets) and it is seen how divisors produce holomorphic line bundles. Two divisors on the complex manifold \(M\) are called equivalent if they differ from a principal divisor (that is, the divisor of a meromorphic function on \(M\)) and equivalent divisors produce isomorphic holomorphic line bundles. A one-to-one map from the set of equivalence classes of divisors into \(Pic(M)\) is given. Chow's theorem states that every analytic set in a complex projective space must be algebraic. Then a non-trivial meromorphic section exists for each holomorphic line bundle and, as a consequence, the above one-to-one map turns out to be an isomorphism. Finally, intersection number on complex surfaces is introduced (the main point is that complex surfaces are real four manifolds, in particular, there is a natural isomorphisms between the rank two De Rham cohomology and rank two homology). Given a complex curve \(S\) in a complex surface \(M\), its Chern's class \(c_{1}([S])\) is then identified with a (class of) two closed form. Then if we are given a holomorphic line bundle \(L\) over \(M\), the intersection of \(S\) and \(L\) is defined by NEWLINE\[NEWLINES \cdot L=\int_{M} c_{1}([S])\wedge c_{1}([L]) = \int_{S} c_{1}(L) \in {\mathbb Z}NEWLINE\]NEWLINE This is used to get a bilinear form \(Div(M) \times Pic(M) \to {\mathbb Z}\) defined by \(D_{1} \cdot D_{2}=D_{1} \cdot [D_{2}]\), where \([D_{2}]\) denotes the holomorphic line bundle associated to the divisor \(D_{2}\). NEWLINENEWLINENEWLINEThe second part of the book starts the main topic of the book. First with holomorphic foliations, with particular interest on complex surfaces and projective spaces. The index of a holomorphic vector field is considered and resolution of singularities studied. The last two chapters restrict to properties of the index in complex surfaces and holomorphic (singular) foliations on compact manifolds. Another book which is not considered in the references [\textit{X. Gomez-Mont} and \textit{L. Ortiz-Bobadilla}, Sistemas Dinámicos Holomorfos en Superficies. Aportaciones Matemáticas No 3, Sociedad matemática Mexicana (1989)] may be considered as a nice complement to this one in the sense that it is written in Spanish and restrict to the case of holomorphic vector fields on complex surfaces.