Algebraic surfaces holomorphically dominable by \(\mathbb{C}^2\)
type: Item
instance of: scholarly article
title: Algebraic surfaces holomorphically dominable by \(\mathbb{C}^2\)
zbMATH Open document ID: 0967.14025
author: Greg Buzzard
author: Steven Lu
published in: Inventiones Mathematicae
publication date: 2000-12-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9903193
review text: An \(n\)-dimensional complex manifold \(M\) is said to be (holomorphically) dominable by \(\mathbb{C}^n\) if there is a map \(F:\mathbb{C}^n\to M\) which is holomorphic such that the Jacobian determinant is not identically zero. Such a map \(F\) is called a dominating map. In this paper, the authors classify algebraic surfaces \(X\) which are dominable by \(\mathbb{C}^2\) using a combination of techniques from algebraic topology, complex geometry and analysis. It is easy to see that if there is a dominating map \(F:\mathbb{C}^2\to X\), then there is also a holomorphic image of \(\mathbb{C}\) which is Zariski dense. One says that an algebraic variety \(X\) satisfies property \(C\) if every holomorphic image of \(\mathbb{C}\) in \(X\) is algebraically degenerate (i.e. is not Zariski dense). For projective surfaces one obtains the following criterion: Theorem 1. Let \(X\) be a projective surface of Kodaira dimension less than 2 and suppose that \(X\) is not birational to a \(K3\) surface which is either elliptic or Kummer. Then \(X\) is dominable by \(\mathbb{C}^2\) if and only if it does not satisfy property \(C\). Then, the following classification is given: Theorem 2. A projective surface \(X\) not birationally equivalent to a \(K3\) surface is dominable by \(\mathbb{C}^2\) if and only if it has Kodaira dimension less than two and its fundamental group is a finite extension of an abelian group (of even rank four or less). If \(k(X)= -\infty\), then the fundamental group condition can be replaced by the simpler condition of non-existence of more than one linearly independent holomorhic 1-form. If \(k(X)=0\) and \(X\) is not birational equivalent to a \(K3\) surface, then \(X\) is dominable by \(\mathbb{C}^2\). If \(X\) is birationally equivalent to an elliptic \(K3\) surface or to a Kummer \(K3\) surface, then \(X\) is dominable by \(\mathbb{C}^2\). More general versions of the above theorems for compact complex surfaces are given at the end of section 4 of the paper…
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 14E99
Mathematics Subject Classification ID: 32J15
zbMATH DE Number: 1443590
zbMATH Keywords: holomorphic maps
zbMATH Keywords: dominating maps
zbMATH Keywords: classification
zbMATH Keywords: holomorphically dominable algebraic surfaces
zbMATH Keywords: \(K3\) surface
zbMATH Keywords: Kodaira dimension
zbMATH Keywords: fundamental group
reviewed by: Vasile Brînzănescu
MaRDI profile type: Publication
OpenAlex ID: W3100279992
arXiv ID: math/9903193
DOI: 10.1007/S002220050021
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description: scientific article