Algebraic surfaces holomorphically dominable by \(\mathbb{C}^2\)
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. Section 5 deals with the non-compact algebraic surfaces, while the last section 6 deals with algebraic surfaces minus small open balls. Many of the tools and results of this paper will be of interest to other areas of mathematics.
holomorphically dominable algebraic surfaces
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