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Algebraic Geometry
Open
Hodge Conjecture
Are Hodge classes on a non-singular complex projective variety algebraic — that is, rational linear combinations of cohomology classes of algebraic subvarieties?
- Impact
- 87 /100
- Funded
- $5,200
- Donors
- 96
- Of goal
- 21%
Why this matters
The Hodge conjecture is a bridge between topology and algebraic geometry. Settling it would clarify which topological cycles can be realized by actual geometric objects.
What a solution unlocks
- Deeper structure theorems for motives and cycles
- New tools relating complex geometry to arithmetic
- Progress on related questions in Hodge theory
Downstream impact
A proof would ripple through algebraic geometry, arithmetic geometry, and the theory of motives — fields that quietly underwrite modern cryptography and coding theory.
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