By Marco Brunella

The textual content offers the birational type of holomorphic foliations of surfaces. It discusses at size the speculation constructed through L.G. Mendes, M. McQuillan and the writer to check foliations of surfaces within the spirit of the class of complicated algebraic surfaces.

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**Extra resources for Birational Geometry of Foliations**

**Sample text**

I=m+1 j=1 This is possible only if all coefficients vanish, because lei, ... , en,+n } is a basis of L. Therefore, the vectors f (e,,,+i , ... , f (em+n) are linearly independent. The theorem is proved. 13. Corollary. The following properties of f are equivalent (in the case of finite-dimension L): a) f is injective, b) dim L = dim Im f . Proof. According to the theorem, dim L = dim Im f , if and only if dim ker f = 0 that is, ker f = {0}. EXERCISES 1. ,n, A. 1. I. ,0) = 0. ' - R", called the differential of f at the point 0, according to the formula (dfo)(ei) _ E a.

En) the basis of the dual space L', and {e...... e;, } the basis of L" dual to {ei, ... , n). ). Actually, CL (e;) is, by definition, a functional on L', whose value at ek is equal to ek(e,) = b:k (the Kronecker delta). But e; is exactly the same functional on L', by definition of a dual basis. We note that if L is infinite-dimensional, then CL : L --+ L** remains injective, but it is no longer surjective (see Exercise 2). In functional analysis instead of the full space L', only the subspaces of linear functionals L' which are continuous in an appropriate topology on L and K are usually studied, and then the mapping L -+ L" can be defined and is sometimes an isomorphism.

Evidently, real Hermitian matrices are symmetric, while anti-Hermitian matrices are antisymmetric. In particular, o(n, R) = u(n) fl sl(n, R). The matrix A is Hermitian if the matrix iA is anti-Hermitian and vice versa. LINEAR ALGEBRA AND GEOMETRY 31 e) The algebra su(n). This is u(n) n sl(n, C) - the algebra of traceless antiHermitian matrices. They form an R-linear space. In Chapter 2, while studying linear spaces equipped with Euclidean or Hermitian metrics, we shall clarify the geometric meaning of operators that are represented by matrices from the classes described above, and we shall also enlarge our list.