真多项式映射的Thom猜想
The Thom Conjecture for proper polynomial mappings
摘要：
设$f,g:X \to Y$是连续映射。我们说$f$拓扑等价于$g$，如果存在同胚$Φ: X\to X$和$Ψ: Y\to Y$使得$Ψ\circ f\circ Φ=g.$设$X,Y$是复光滑不可约仿射簇.证明了多项式映射的代数族$F: M\times X\ni (m, x)\mapsto F(m, x)=f_m(x)\in Y$只包含有限个拓扑不等价的真映射。特别是只有有限数量的拓扑不等价的正常多项式映射$f: \Bbb C^n\to\Bbb C^m$的有界（代数）度。这在真多项式映射的情况下给出了Thom猜想的一个肯定答案。
英文摘要：
Let $f,g:X \to Y$ be continuous mappings. We say that $f$ is topologically equivalent to $g$ if there exist homeomorphisms $Φ: X\to X$ and $Ψ: Y\to Y$ such that $Ψ\circ f\circ Φ=g.$ 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.
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