Fano-Mukai fourfolds of genus 10 and their automorphism groups
Mikhail Zaidenberg
(Institut Fourier, Grenoble)

The celebrated Hirzebruch Problem asks to describe all possible smooth compactifications of C^n  with second Betti number 1. Projective completions of C^n  $  are Fano varieties; in dimension at most 3 they are all known (Remmert-van de Ven, Brenton-Morrow, Peternell, Prokhorov, Furushima). It occurs that any variety in the title provides a new example in dimension 4. These varieties form a 1-parameter family. The group Aut^0(V) of a general member V of this family is  isomorphic to the algebraic 2-torus (C^*)^2. There are two exceptional members of the family with ${\rm Aut}^0(V)$ equal GL(2, C} and C x C^*, respectively. The discrete part of the automorphism group Aut(V) is a finite cyclic group. To compute Aut(V) we use three different geometric realizations of Aut(V).  The talk is based on a joint work with Yuri Prokhorov.