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On Chern numbers of homology planes of certain types. (English) Zbl 0891.14007

A homology plane is a nonsingular algebraic surface \(X\) defined over \(\mathbb{C}\) with \(H_i(X; \mathbb{Z})=0\) for \(i>0\). In this case, \(X\) is an affine rational surface and has a fiber space structure whose general fibers are isomorphic to \(\mathbb{C}^{N*}:=\) the affine line minus \(N\) points. The case \(N=1\) corresponds to Kodaira dimension \(\leq 1\). The authors are interested in the following problem: Does there exist a constant \(A\) such that, for every homology plane \(X\), \(X\) admits a \(\mathbb{C}^{N*}\)-fibration for some \(N\leq A\)? For an open surface \(X\), one can define the logarithmic Chern numbers \(c^2_1\) and \(c_2\) (in this case, \(c^2_1\) could be a rational number) and the Miyaoka-Yau inequality \(c^2_1\leq 3c_2\) still holds. If \(X\) is a homology plane then \(c_2=1\).
The authors prove that, for a homology plane \(X\) of general type with a \(\mathbb{C}^{**}\)-fibration [such homology planes were classified by M. Miyanishi and T. Sugie in Osaka J. Math. 28, No. 1, 1-26 (1991; Zbl 0749.14029)], one has \(c^2_1<2\) and there exists a sequence of such homology planes with \(c^2_1\) converging to 2. The authors also calculate \(c^2_1\) for the homology planes of general type with a \(\mathbb{C}^{3*}\)-fibration constructed by T. tom Dieck (1992). It turns out that, in this case, \(c^2_1<{5\over 2}\) and there is a sequence of such homology planes with \(c^2_1\) converging to \(5\over 2\). These computations lead the authors to ask the following question: Does there exist a sequence of homology planes with \(c^2_1\) converging to 3? If such a sequence would exist, then it is more plausible that the answer to the problem formulated at the beginning is negative.

MSC:

14F45 Topological properties in algebraic geometry
57R20 Characteristic classes and numbers in differential topology

Citations:

Zbl 0749.14029
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