Hodge conjecture

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Hodge conjecture

[′häj kən‚jek·chər]

(mathematics)

The 2 p-dimensional rational cohomology classes in an n-dimensional algebraic manifold M which are carried by algebraic cycles are those with dual cohomology classes representable by differential forms of bidegree (n-p, n-p) on M.

References in periodicals archive ?

For very general Z, it is known that [N.sup.2][H.sup.4](X) (which is the subspace of Hodge classes, as the Hodge conjecture is known for X) has dimension 2.

A remark on the Chow ring of some hyperkahler fourfolds

Heck, how can you expect to solve the "Hodge conjecture" without it?

Rational coefficients

Specifically, they are the Riemann hypothesis, which lingers from Hilbert's list, Yang-Mills theory and the mass gap hypothesis, the P Versus NP problem, the Navier-Stokes equations, the Poincaire conjecture, the Birch and Swinnerton-Dyer conjecture, and the Hodge conjecture. Devlin profiles each problem and offers insight into how it came about and its significance.

The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time

Professor Miles Reid, of Warwick University, has spent more than 30 years mulling over the Hodge Conjecture which has baffled academics for decades.

But Prof Reid, 52, claims the Hodge Conjecture will never be solved and has instead channelled his efforts into disproving the complex algebraic theory.