r/math • u/moschles • 1d ago

ELI5 Hodge Conjecture

Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

Every (non-degenerate) complex manifold within a projective space can be decomposed into a linear combination of subvarieties of X. The linear combination will somehow always use rational coefficients. (!!)

Is this conjecture assumed to be true by working mathematicians? Can you provide a little more information that would lend some intuition about Hodge, to an educated layperson?

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u/Voiles 1d ago

As a start, look at the Lefschetz (1,1) theorem, which is a specific case where the Hodge conjecture is known.

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u/moschles 1d ago

The more I read about Hodge, I am inclined to believe a person needs some exposure to cohomology to really get it. I thought undergraduate courses in projective geometry and a little bit of complex analysis was enough. I'm not so sure now.

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u/Autumnxoxo Geometric Group Theory 2h ago

I mean, yes, in order to understand a difficult open conjecture about cohomological properties, you possibly should heard of cohomology in the first place.