Hodge conjecture

Hodge conjecture is the algebraic topology of a non-singular complex algebraic variety. Therefore, elliptic curves should be correlated with it because of the rational point.

X is a compact complex manifold of complex dimension n. Moreover, its cohomology groups lie in degrees zero through 2n. Elliptic curves are the two dimensions, although you pile them in Riemann sphere. There must be a decomposition on its cohomology with complex coefficients. (p,q) is harmonic forms. These are the cohomology classes represented by differential forms such as Z1,...,Zn.

Z is a complex submanifold of X.

i:Z→X

α is integration.

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