The Thom Conjecture for proper polynomial mappings

TL;DR

This paper proves that within algebraic families of polynomial mappings between complex affine varieties, only finitely many are topologically non-equivalent, confirming the Thom Conjecture for proper polynomial mappings.

Contribution

It establishes finiteness of topologically non-equivalent proper polynomial mappings in algebraic families, affirming the Thom Conjecture for these mappings.

Findings

Finite number of topologically non-equivalent proper mappings in algebraic families.

Finiteness of proper polynomial mappings of bounded degree.

Positive resolution of 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 $f: \Bbb C^n\to\Bbb C^m$ of bounded (algebraic) degree. This gives a positive answer to the Thom Conjecture in the case of proper polynomial mappings.