On local invariants of totally real surfaces

  • G. KHIMSHIASHVILI
  • Published 2003

Abstract

We will be basically concerned with germs of totally real two-dimensional surfaces with isolated singularities in C given by polynomial parametrisations. The ultimate goal we have in mind, is to develop some tools which could help to derive information about the geometric properties of generic deformations of such germs (e.g., existence of complex tangents [4] and selfintersections [6]) from the algebraic data of parameterizing polynomials. In particular, we would like to obtain some effectively computable estimates for possible values of their (local) topological invariants. In the present paper we realize this program for a natural topological invariant of such germs, called self-intersection index, which was introduced in [17] (see also [7], [10], [13]). For simplicity, we will only deal with germs of totally real surfaces in C despite our approach is apparently applicable to totally real surfaces in arbitrary almost complex four-dimensional manifolds. Main attention will be given to germs given by quasi-homogeneous parametrisations. Such germs regularly appear in many aspects of symplectic geometry [7] and complex analysis [17], [10] (in fact, all examples considered in [7], [17], [10] belong to this class) so they may deserve some attention on themselves. The main result (Theorem 2) yields an upper estimate for the absolute value of self-intersection index in terms of weights and degrees of parameterizing quasi-homogeneous polynomials. Its proof relies on an algebraic method of computing self-intersection indices by means of so-called signature formulas for the topological degree [5], [11] which involve rather specific algebraic notions and constructions. In order to make the exposition more clear and self-contained, we describe this algebraic technique in some detail and then explain how to use it for our purposes. We start by discussing indices (topological degrees) of polynomial vector fields [3]. In particular, we obtain an algebraic estimate for the index of a quasi-homogeneous vector field (Theorem 1) which generalizes similar results

Cite this paper

@inproceedings{KHIMSHIASHVILI2003OnLI, title={On local invariants of totally real surfaces}, author={G. KHIMSHIASHVILI}, year={2003} }