\(\mathbb Q\)-homology planes with \(\mathbb C^{**}\)-fibrations
Let \(X/\mathbb C\) be a nonsingular algebraic surface. We call \(X\) a homology plane (resp. a \(\mathbb Q\)-homology plane) if \(H_i(X;\mathbb Z)=(0)\) (resp. \(H_i(X;\mathbb Q)=(0))\) for \(1\leq i\leq 4\). Similarly, a logarithmic homology plane is a normal affine surface with at worst quotient singularities satisfying \(H_i(X;\mathbb Z)=(0)\) for \(i\geq 1\). A homology plane with the logarithmic Kodaira dimension \(\overline\kappa\leq 1\) was already studied and classified. Especially, homology planes with \(\overline\kappa=1\) are known to admit a \(\mathbb C^*\)-fibration. As a natural extension of the \(\mathbb C^*\)-fibrations, in the paper, we study logarithmic \(\mathbb Q\)-homology planes which admit a \(\mathbb C^{**}\)-fibration, where \(\mathbb C^{**}\) denotes \(\mathbb P^1\) minus three points. We classify possible singular fibers of such fibration and calculate the order of \(H_1(X;\mathbb Q)\). As a result, we can describe all \(\mathbb Z\)-homology planes and logarithmic \(\mathbb Z\)-homology planes with \(\overline\kappa=2\) having a \(\mathbb C^{**}\)-fibration. Also we get infinitely many \(\mathbb Z\)-homology planes with \(\overline\kappa=2\) which have a nontrivial finite group of automorphisms.
scientific article; zbMATH DE number 9160