A connection between automorphism groups of K3 surfaces over the field of complex numbers $\mathbb C$ and groups generated by reflections in Lobachevsky spaces first appeard in a classical work of Piatetski-Shapiro and Shafarevich in which they established a global Torelli theorem for K3 surfaces. In particular, they showed that the automorphism group of a K3-surface over $\mathbb C$ is finite if and only if the automorphism groups of its Picard lattice is generated up to finite index by reflections in its elemets of square $(-2)$. The Picard lattice is hyperbolic over $\mathbb Z$ and determines a Lobachevsky space on which the group generated by reflections in elemets of square $(-2)$ acts discretely. Shortly before, Vinberg and Makarov obtained important results about arithmetic groups generated by reflections in Lobachevsky spaces. Both groups of authors attempted to use this connection in order to describe automorphism groups of K3 surfaces and arithmetic groups generated by reflections in Lobachevsky spaces. Important results in this direction were obtained by Nikulin around 1980, that is, 10 years later.

There has been a lot of progress in this area, which is the topic of the course.

Просьба к участникам обращаться к Вячеславу Валентиновичу Никулину, nikulin@mi-ras.ru, за данными для подключения к занятиям через Zoom.

Financial support. The course is supported by the Simons Foundation and the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2019-1614).

Lecturer
Nikulin Viacheslav Valentinovich

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center