$\mathbb{A}^1_*$-fibrations on affine threefolds

This is a joint work with R.V. Gurjar, M. Koras, K. Masuda and P. Russell. In affine algebraic geometry, our knowledge on affine algebraic surfaces is fairly rich with various methods of studying them. Meanwhile, knowledge of affine threefolds is very limited. It is partly because strong geometric approaches are not available or still under development. A possible geometric approach is to limit ourselves to the case where the affine threefolds in consideration have fibrations by surfaces or curves, say the affine plane $\mathbb{A}^2$ or the curves $\mathbb{A}^1$ and $\mathbb{A}^1_*$, or have group actions of $\mathbb{G}_a$ or $\mathbb{G}_m$ which give the quotient morphisms over affine surfaces. Here the symbol $\mathbb{A}^1_*$ signifies the affine line $\mathbb{A}^1$ minus one point, often written as $\mathbb{C}^*$. The main theme of the present talk is $\mathbb{A}^1_*$-fibrations defined on affine threefolds. The difference between $\mathbb{A}^1_*$-fibration and the quotient morphism by a $\mathbb{G}_m$-action is more essential than in the case of an $\mathbb{A}^1$-fibration and the quotient morphism by a $\mathbb{G}_a$-action. We consider necessary (and partly sufficient) conditions using the types of singular fibers under which a given $\mathbb{A}^1_*$-fibration becomes the quotient morphism by a $\mathbb{G}_m$-action. The structure of such affine threefolds can be elucidated. We also consider homology (or $\mathbb{Q}$-homology, or contractible) threefolds with $\mathbb{A}^1_*$-fibrations and apply the results to these classes of threefolds. For example, we observe flat $\mathbb{A}^1_*$-fibrations which are expected to be surjective, but this turns out to be not the case by an example of Winkelmann. This example gives also a quasi-finite endomorphism of $\mathbb{A}^2$ which is not surjective.