Complex hypersurfaces diffeomorphic to affine spaces

Simple birational extensions of the polynomial algebra ℂ^{[3]}

“…A complete analog of Theorem 7.2 in [19] cited at the beginning holds if the polynomial p ∈ C [4] is linear with respect to two (and not just one) variables. Indeed (3.24) if the 3-fold X, p = a(x, y)u + b(x, y)v + c(x, y) = 0, in C 4 is smooth and acyclic, then p ∈ C [4] is a variable.…”

“…The main results 4.2, 4.16 of subsection 4.1 provide sufficient conditions for when an x-residual variable p = f u + g ∈ C [4] as in (1) is indeed a variable. For instance (see 4.2, 4.3) this is the case if deg z g ≤ 1, or f is a power of an irreducible polynomial, or else f ∈ C[x] (the latter result strengthens those of M. Miyanishi [28,Thm.…”

“…Indeed (3.24) if the 3-fold X, p = a(x, y)u + b(x, y)v + c(x, y) = 0, in C 4 is smooth and acyclic, then p ∈ C [4] is a variable. We give a simple criterion (in terms of the coefficients a, b, c ∈ C[x, y]) for when this is the case.…”

“…If furthermore (3.21) X is isomorphic to C 3 , then any fiber X λ := p −1 (λ) (λ ∈ C) of the polynomial p is isomorphic to C 3 as well, and moreover (with an appropriate choice of coordinates (x, y)) all fibers of the morphism are reduced and isomorphic to C 2 . We do not know whether in that case a polynomial p in (1) must be a variable of the polynomial ring C [4] , and ρ must be a trivial family. However, in section 4 in many cases we provide affirmative answers to these questions and give simple concrete examples where the answers remain unknown.…”

“…In 2.28, 3.21 and 3.6 below we give a criterion for when a 3-fold X = p −1 (0) ⊂ C 4 with p = f (x, y)u + g(x, y, z) ∈ C [4] (f ∈ C [2] \{0}, g ∈ C [3] ) (1) is acyclic (resp., is isomorphic to C 3 , resp., is an exotic C 3 ). In particular, we show in 2.11 that if X is acyclic, then actually it is diffeomorphic to R 6 .…”