The Hodge conjecture for self-products of certain K3 surfaces

TL;DR

This paper proves the Hodge conjecture for self-products of specific K3 surfaces, using endomorphism algebra analysis of associated Kuga--Satake varieties and real multiplication properties.

Contribution

It applies van Geemen's result to determine endomorphism algebras and proves the Hodge conjecture for certain double cover K3 surfaces.

Findings

Hodge conjecture verified for these K3 surfaces

Endomorphism algebra characterized for Kuga--Satake varieties

Application of real multiplication in Hodge theory

Abstract

We use a result of van Geemen to determine the endomorphism algebra of the Kuga--Satake variety of a K3 surface with real multiplication. This is applied to prove the Hodge conjecture for self-products of double covers of $\PP^2$ which are ramified along six lines.