SMF

An analytic cancellation theorem and exotic algebraic structures on $\mathbb {C}^n$

An analytic cancellation theorem and exotic algebraic structures on $\mathbb {C}^n$

M. ZAINDENBERG
  • Année : 1993
  • Tome : 217
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
    Anglais
  • Pages : 251-282
  • DOI : 10.24033/ast.236

A construction of a series $\{X_i\}_{i=1,2,\ldots }$ of topologically contractible smooth complex affine algebraic surfaces of log-general type is presented. By an idea due to C.P. Ramanujam (1971), for each $n \geq 3$ this gives a series of exotic $\mathbb {C}^n$, the affine manifolds $X_i \times \mathbb {C}^{n-2}$ diffeomorphic but not biholomorphic to $\mathbb {C}^n$. The following analytic concellation theorem ensures that these exotic algebraic structures are analytically different : Given a biholomorphic $X \times \mathbb {C}^k \to Y \times \mathbb {C}^k$ where $X$ and $Y$ are quasiprojective varieties of log-general type, the factor $\mathbb {C}^k$ can be cancelled, resulting with a biregular isomorphism $X \to Y$. It is also shown that none of the above exotic $\mathbb {C}^n$ contains a hypersurface which is the image of $\mathbb {C}^{n-1}$ under a regular injection.

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