The Open Mathematics Journal
2008, 1 : 1-10 Published online 2008 May 6. DOI: 10.2174/1874117400801010001
Publisher ID: TOMATJ-1-1
On the Poincar´e-Hopf Index Theorem for the complex case
Instituto de Matem´atica - Universidade Federal do Rio
de Janeiro - Caixa Postal 68530-Rio de Janeiro-RJ, 21.945-970, Brasil.
ABSTRACT
In this paper we consider holomorphic vector fields and holomorphic one-forms defining distributions which are transverse to the boundary of a regular domain on a complex manifold or on the complex affine space. In both cases we prove that, under a natural extensibility topological hypothesis, the Euler-Poincar´e-Hopf characteristic of the domain is equal to the sum of indexes of the complex vector field or complex one-form at its singular points inside the domain.