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).
RSS: Forthcoming seminars
Lecturer
Nikulin Viacheslav Valentinovich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow Steklov International Mathematical Center |