Zbigniew Jelonek (Instytut Matematyczny,)

Tuesday, May 23th, 10:20, Room 4-111

Title: On Finite Regular and Holomorphic Mappings

Abstract:  

Let X, X', Y be smooth algebraic varieties of the same dimension. Let f : X → Y , g : X' → Y be finite regular mappings. We say that f, g are equivalent

if there exists a regular isomorphism Φ : X → X' such that f = g ◦ Φ. We show that for every hypersurface V ⊂ Y and every k ∈ N, there are only a finite number of non-equivalent finite regular mappings f : X → Y such that the discriminant D(f) equals V and μ(f) = k. In particular if Krn = {x ∈ Cn:Qri=1 xi = 0} and X is a smooth and simply connected algebraic manifold, then every finite regular mapping f : X → Cn with D(f) = Krn is equivalent to one of the mappings

 fd1,...,dr: Cn  (x1, . . . , xn) → (xd11, . . . , 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) is a proper and holomorphic mapping of topological degree two, then there exist biholomorphisms Ψ, Φ : (Cn, 0) → (Cn, 0) such that 

 Ψ ◦ f ◦ Φ(x1, x2, . . . , xn) = (x21, x2, . . . , xn). Moreover, for every proper holomorphic mapping f : (Cn, 0) → (Cn, 0) which has a discriminant with only simple normal crossings, there exist biholomorphisms 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.