Dmitry Timashev. When the automorphism group of a projective variety is linear algebraic?

Семинар лаборатории алгебраических групп преобразований, https://cs.hse.ru/latg/seminar

Дата: 01.04.2026

Докладчик: Dmitry Timashev

Тема: When the automorphism group of a projective variety is linear algebraic?

Аннотация: This talk may be regarded as a continuation of my talk on 15-10-2025. As we have seen in that talk, the automorphism group of a projective (even complete) algebraic variety is represented by a group scheme of locally finite type whose identity component is an algebraic group (the Matsumura-Oort theorem, 1967). However the connected automorphism group Aut^0(X) of a projective variety X may be quite far from linear. For instance, if X is not uniruled, then Aut^0(X) is an Abelian variety. In fact, in characteristic 0 any connected algebraic group can be realized as
Aut^0(X) for some smooth projective variety X (Brion, 2014).
It is an interesting question under which conditions the group Aut^0(X) or even the whole automorphism group Aut(X) is linear algebraic. We shall discuss several necessary or sufficient conditions for that and consider examples. In particular, Aut^0(X) is a linear algebraic group if the Picard group Pic(X) is discrete, and Aut(X) is linear algebraic if X is Fano or equipped with a locally transitive action of a linear algebraic group. (The latter result is due to Fu-Zhang, 2013, in the complex analytic setting and to Brion, 2018, in the algebraic setting.) On the other side, for varieties of general type, Aut(X) is finite.
In our exposition, we mostly follow the "Notes on automorphism groups of projective varieties" by M. Brion (2018, https://www-fourier.univ-grenoble-alpes.fr/~mbrion/autos_final.pdf).

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