*This article is part of a series on Complex Algebraic Geometry. See also:*

1. The Hardest Conjecture; 2. The Valley; 3. A Mathematical Daydream

“Those limits in certain functor categories that cannot be computed pointwise? They don’t actually exist,” one of us declared authoritatively. We, all math grad students, sat on an apartment floor, in a circle, with cards scattered over the floor.

Blank looks abounded. “Meaning, there don’t actually exist monic natural transformations some of whose components are not monomorphisms,” he clarified.

We had invented a variant of a popular card game called “spies vs. revolutionaries” – we called it “students vs. professors” – in which, in particular, each new round was heralded by its leader’s presentation of an established mathematical truth that he or she had decided we were to overturn for good. “The Banach-Tarski paradox is still true, but requires using at best six pieces, not five,” another student later suggested. “2 isn’t actually a prime,” one student blustered, citing the integer’s pathological character in many number-theoretic environments.

Soon it was my turn. “The Hodge Conjecture is false,” I fibbed, “and a counter-example is provided in my paper.”

A good laugh and a general readiness to proceed with the game cut my monologue short. Ridiculous as it was, though, I was ready to continue. The matter was one to which I had given some thought. Indeed, my work presents an interesting testing ground for a few of the ideas surrounding the Hodge Conjecture, and in particular seems to invite a heuristic argument whereby it could be used to furnish a counter-example to the conjecture. I’ll explain this mathematical daydream, and how it can be ultimately debunked. Continue reading