Equivariant birational geometry of rational varieties for finite group actions

Abstract

It is nowadays well known that the finite subgroups of the Cremona group are isomorphic to the finite groups that act biregularly on rational varieties. While the question of classifying these subgroups of Cremona up to isomorphism would be answered by finding all the groups that can act on rational varieties, one can ask for a finer classification, namely up to conjugation. This leads to the beautiful theory of G-equivariant birational geometry of rational varieties.

We can outline two particularly important questions. The first is the linearization problem, that is, determining whether a subgroup G of Crn(C) is conjugate to a subgroup of Aut(Pn), or equivalently, the G-equivariant rationality problem. The second is the G-solidity problem. Roughly speaking, a variety is G-solid if G is not conjugate in Crn(C) to a group that can be decomposed into subgroups of Cremona groups of smaller rank.

In dimension two, both questions remained open after the seminal works of Blanc and Dolgachev– Iskovskikh, and we completely answer them in the present thesis. In dimension three, we further the investigations of Cheltsov–Shramov and Prokhorov on actions of the icosahedral group A5 on Fano threefolds, and we treat the surprisingly open case of smooth quadrics. We provide all the G-birational models for the fixed-point-free actions of the icosahedral group A5 on these varieties, and we answer the questions of linearizability and solidity for all such actions.

Lastly, we became interested in the Fano threefolds obtained by blowing up a non-hyperelliptic curve of degree six and genus three in the projective space, because their automorphism groups beautifully arise from actions on smooth plane quartics. We construct all the possible isomorphism classes of groups acting faithfully on such a variety. These threefolds also provided an opportunity to dive into the world of K-stability. In the final part of this thesis, we present how we linked their G-equivariant geometry to the existence of a Kähler–Einstein metric and produced many K-stable examples.

We will essentially consider the actions of finite groups, and, unless stated otherwise, all the work will be done over the field of complex numbers.