On finite regular and holomorphic mappings

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• 作者：Zbigniew Jelonek ^{najelone@cyf-kr.edu.pl}
• 关键词：14A10 ; 14R10 ; 51M99
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：14 January 2017
• 年：2017
• 卷：306
• 期：Complete
• 页码：1377-1391
• 全文大小：367 K
• 卷排序：306

文摘

We show that for every hypersurface V⊂YV⊂Y and every k∈Nk∈N, there are only a finite number of non-equivalent finite regular mappings f:X→Yf:X→Y such that the discriminant D(f)D(f) equals V   and μ(f)=kμ(f)=k. In particular if Knr={x∈Cn:∏i=1rxi=0} and X   is a smooth and simply connected algebraic manifold, then every finite regular mapping f:X→Cnf:X→Cn with D(f)=Knr is equivalent to one of the mappings fd1,…,dr:Cn∋(x1,…,xn)↦(x1d1,…,xrdr,xr+1,…,xn)∈Cn. Moreover, we obtain generalizations of the Lamy Theorem. We prove the same statement in the local (and sometimes global) holomorphic situation. In particular we show that if f:(Cn,0)→(Cn,0)f:(Cn,0)→(Cn,0) is a proper and holomorphic mapping of topological degree two, then there exist biholomorphisms Ψ,Φ:(Cn,0)→(Cn,0)Ψ,Φ:(Cn,0)→(Cn,0) such that Ψ∘f∘Φ(x1,x2,…,xn)=(x12,x2,…,xn). Moreover, for every proper holomorphic mapping f:(Cn,0)→(Cn,0)f:(Cn,0)→(Cn,0) which has a discriminant with only simple normal crossings, there exist biholomorphisms Ψ,Φ:(Cn,0)→(Cn,0)Ψ,Φ:(Cn,0)→(Cn,0) such that Ψ∘f∘Φ(x1,x2,…,xn)=(x1d1,x2d2,…,xrdr,xr+1,…,xn), where r is the number of irreducible components of the discriminant at 0.