Counterexample to the Hodge Conjecture

2006/08/10 by Kim, K. H., Roush, F. W.
#14C30 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0608265

Abstract

We construct a K3 surface whose transcendental lattice has a self-isomorphism which is not a linear combination of self-isomorphisms over 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.

Related

• The Hodge conjecture for self-products of certain K3 surfaces
• The Hodge standard conjecture for self-products of K3 surfaces
• On the Hodge conjecture for products of certain surfaces
• Hodge similarities, algebraic classes, and Kuga-Satake varieties
• Kuga-Satake varieties and the Hodge conjecture