On Vorontsov's theorem on K3 surfaces

TL;DR

This paper proves Vorontsov's theorem on K3 surfaces, focusing on the relationship between automorphism groups, the transcendental lattice, and the classification of these surfaces, especially in the non-unimodular case.

Contribution

It completes the proof of Vorontsov's theorem for K3 surfaces by addressing the non-unimodular transcendental lattice case, extending previous results.

Findings

Confirmed the classification of K3 surfaces with specific lattice properties.

Extended Vorontsov's theorem to non-unimodular cases.

Provided explicit realizations for the classified surfaces.

Abstract

Let X be a K3 surface with the Neron-Severi lattice S_X and transcendental lattice T_X. Nukulin considered the kernel H_X of the natural representation Aut(X) ---> O(S_X) and proved that H_{X} is a finite cyclic group with phi(h(X))) | t(X) and acts faithfully on the space H^{2,0}(X) = C omega_{X}, where h(X) = ord(H_X), t(X) = rank T_X and phi(.) is the Euler function. Consider the extremal case where phi(h(X)) = t(X). In the situation where T_{X} is unimodular, Kondo has determined the list of t(X), as well as the actual realizations, and showed that t(X) alone uniquely determines the isomorphism class of X (with phi(h(X)) = t(X)). We settle the remaining situation where T_X is not unimodular. Together, we provide the proof for the theorem announced by Vorontsov.