In mathematics, the Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the homology classes of subvarieties.
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- This 'Hodge conjecture' has by now achieved a considerable status, almost on a par with the Riemann hypothesis or the Poincaré conjecture.
- Michael Atiyah (28 April 1988). Collected Works: Michael Atiyah Collected Works: Volume 1: Early Papers; General Papers. Clarendon Press. pp. 250. ISBN 978-0-19-853275-0.
- The twin conjectures of Hodge and Tate have a status in algebraic and arithmetic geometry similar to that of the Riemann hypothesis in analytic number theory.
- The Hodge conjecture postulates a deep and powerful connection between three of the pillars of modern mathematics: algebra, topology, and analysis.
- Ian Stewart (5 March 2013). Visions of Infinity: The Great Mathematical Problems. Basic Books. p. 211. ISBN 978-0-465-06599-8.
- Riemann hypothesis
- P versus NP problem
- Poincaré conjecture (solved)
- Birch and Swinnerton-Dyer conjecture