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Lefschetz Standard Conjecture

The Lefschetz standard conjecture is the assertion that certain inverse isomorphisms on cohomology are induced by algebraic cycles. More explicitly, let X be a smooth complex projective variety of complex dimension d, and let h in H^2(X,Q) be the cohomology class of a hyperplane section. For 0<=i<=d, the cup product map

L^(d-i):H^i(X,Q)->H^(2d-i)(X,Q),

where L^(d-i)(alpha)=h^(d-i) union alpha, is an isomorphism. The conjecture requires its inverse to be induced by an algebraic cycle with rational coefficients on X×X (Kleiman 1994).

For a projective algebraic variety deformation equivalent to a generalized Kummer variety of positive complex dimension divisible by four, this conjecture implies the Hodge conjecture for the variety and all its self-powers (Floccari and Fu 2026).

See also

Algebraic Cycle, Cup Product, Hodge Conjecture, Projective Variety

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References

Floccari, S. and Fu, L. "On the Hodge Conjecture for Hyper-Kähler Varieties of Generalized Kummer Type of Dimension Divisible by Four." 22 Oct 2026. https://arxiv.org/abs/2610.31740.Kleiman, S. L. "The Standard Conjectures." In Motives, Part I (Ed. U. Jannsen, S. L. Kleiman, and J.-P. Serre). Providence, RI: Amer. Math. Soc., pp. 3-20, 1994. https://doi.org/10.1090/pspum/055.1/1265519.

Cite this as:

Weisstein, Eric W. "Lefschetz Standard Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LefschetzStandardConjecture.html

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