A paper appeared last night on the arXiv by K.H. Kim and F.W. Roush entitled Counterexample to the Hodge Conjecture. The authors claim to construct an example using K3 surfaces for which the Hodge conjecture is false. If they’re right about this, this would be very shocking, and I would guess that most experts will be very skeptical about the result. Most likely someone soon will find a problem with the argument, but if not there will be a lot of excitement.
The Hodge conjecture is one of the Clay Millenium prize problems, so if this paper is right, the authors may very well be entitled to $1 million. For more about what the Hodge conjecture says, see the slides or video of a popular lecture by Dan Freed, or the official statement of the problem due to Pierre Deligne.
Update: The authors have withdrawn their claim to have disproven the Hodge conjecture, acknowledging problems with their argument beginning in section 5 of the paper.
Last Updated on