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What is Hodge Conjecture |
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Answer» Answer: Step-by-step explanation: LET: We CALL this the group of Hodge classes of degree 2k on X. The modern statement of the Hodge conjecture is: 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. A projective complex manifold is a complex manifold which can be embedded in complex projective space. Because projective space carries a Kähler metric, the Fubini–Study metric, such a manifold is always a Kähler manifold. By Chow's theorem, a projective complex manifold is ALSO a smooth projective algebraic variety, that is, it is the ZERO set of a collection of homogeneous polynomials. |
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