The Thom Conjecture for proper polynomial mappings

Abstract

Let $f,g:X\to Y$ be continuous mappings. We say that $f$ is topologically equivalent to $g$ if there exist homeomorphisms $\Phi :X\to X$ and $\Psi :Y\to Y$ such that $\Psi \circ f\circ \Phi =g.$ Let $X,Y$ be complex smooth irreducible affine varieties. We show that every algebraic family $F:M\times X\ni (m,x)\mapsto F(m,x)=f_m(x)\in Y$ of polynomial mappings contains only a finite number of topologically non-equivalent proper mappings. In particular there are only a finite number of topologically non-equivalent proper polynomial mappings $f:{\mathbb{C}}^n\to {\mathbb{C}}^m$ of bounded (algebraic) degree. This gives a positive answer to the Thom Conjecture in the case of proper polynomial mappings.