Location:

Seminar room 00.003

## Seminar Algebra and Geometry: Pierre-Marie Poloni (Bern)

**Some affine plane bundles over the punctured affine plane**

An $\mathbb{A}^2$-fibration is a flat morphism between complex affine varieties whose fibers are isomorphic to the complex affine plane. In this talk, we study explicit families $f:\mathbb{A}^4\to\mathbb{A}^2$ of $\mathbb{A}^2$-fibrations over the affine plane.

The famous Dolgachev-Weisfeiler conjecture predicts that such fibrations are in fact isomorphic to the trivial bundle. We will show that this holds true in some particular examples. For instance, we will recover a result of Drew Lewis which states that the $\mathbb{A}^2$-fibration induced by the second Vénéreau polynomial is trivial.

Our proof is inspired by a previous work of Kaliman and Zaidenberg and consists in first showing that the considered fibrations have a fiber bundle structure when restricted over the punctured affine plane.

This is a joint work in progress with Jérémy Blanc.

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