Counterexample to the Hodge Conjecture

Abstract

We construct a K3 surface whose transcendental lattice has a self-isomorphism which is not a linear combination of self-isomorphisms over $\mathbb{Q}$ which preserve cup products up to nonzero multiples. Products of it with itself give candidates for counterexamples to the Hodge conjecture which may be of interest. We also study the Kuga-Satake construction in relation to these examples.

We no longer claim to have disproved the Hodge conjecture. Section 5 is deleted except for 5.3, and Theorems 6.2,6.3 are deleted

References in corpus (2)

• Vector bundles on a K3 surface
• Complex Multiplication for K3 Surfaces