Algebraic GeometryUnsolved
Hodge Conjecture
Are certain cohomology classes on smooth projective varieties representable by algebraic cycles?
Overview
The Hodge Conjecture concerns the relationship between algebraic cycles and certain cohomology classes on smooth projective complex varieties.
Why It Matters
It connects geometry, topology, and algebraic geometry at a deep level.
Current Status
Proven in special cases but remains open in general.
What a Solution Would Require
A general proof (or counterexample) showing whether all rational Hodge classes are algebraic cycles.