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Decomposition of the Diagonal

A decomposition of the diagonal is a Chow ring identity expressing the diagonal of a smooth projective algebraic variety X of dimension n, over an algebraically closed field, as

[Delta_X]=[X×{x}]+[Z],

where x in X and Z is an algebraic cycle of dimension n supported on D×X for a proper closed set D subset X. The equality is in CH_n(X×X) with integer coefficients, modulo rational equivalence, not merely in cohomology or with rational coefficients (Voisin 2017).

For smooth projective varieties over an algebraically closed field, being a rational variety, a stably rational variety, or a retract rational variety implies a decomposition of the diagonal (Hotchkiss 2026). Therefore, failure of such a decomposition obstructs all three properties.

See also

Algebraic Cycle, Algebraic Variety, Chow Ring, Hypersurface, Rational Variety, Retract Rational Variety, Stably Rational Variety

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References

Hotchkiss, J. "The Rationality Problem for Hypersurfaces of Degree at Least Five." 7 Oct 2026. https://arxiv.org/abs/2610.10509.Voisin, C. "On the Universal CH_0 Group of Cubic Hypersurfaces." J. Eur. Math. Soc. 19, 1619-1653, 2017. https://doi.org/10.4171/JEMS/702.

Cite this as:

Weisstein, Eric W. "Decomposition of the Diagonal." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DecompositionoftheDiagonal.html

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