Math Conjectures

Lefschetz Standard Conjecture BB

OPENMajorConjectureProposed 1968 · Standard version

Canonical statement

Let XX be a smooth projective variety of dimension dd over an algebraically closed field, let H∗H^\ast be a Weil cohomology theory with characteristic-00 coefficient field, and let L(α)=h⌣αL(\alpha)=h\smile\alpha for the class hh of an ample divisor. For every i<di<d, the inverse of the hard-Lefschetz isomorphism
Ld−i:Hi(X)→∼H2d−i(X) L^{d-i}:H^i(X)\xrightarrow{\sim}H^{2d-i}(X)
is induced by an algebraic correspondence in CH⁡i(X×X)⊗Q\operatorname{CH}^i(X\times X)\otimes\mathbb Q, where CH⁡i\operatorname{CH}^i denotes codimension-ii algebraic cycles modulo rational equivalence.
View source LaTeX
Let \(X\) be a smooth projective variety of dimension \(d\) over an algebraically closed field, let \(H^\ast\) be a Weil cohomology theory with characteristic-\(0\) coefficient field, and let \(L(\alpha)=h\smile\alpha\) for the class \(h\) of an ample divisor. For every \(i<d\), the inverse of the hard-Lefschetz isomorphism
\[
  L^{d-i}:H^i(X)\xrightarrow{\sim}H^{2d-i}(X)
\] is induced by an algebraic correspondence in \(\operatorname{CH}^i(X\times X)\otimes\mathbb Q\), where \(\operatorname{CH}^i\) denotes codimension-\(i\) algebraic cycles modulo rational equivalence.
AG-CYCLE-004:[Grothendieck1969Standard][Kleiman1968Standard][Andre2004Motives]
Algebraic geometryAlgebra and representation theory

Notes

Let XX be a smooth projective variety of dimension dd with a Weil cohomology theory H∗H^\ast, and let LL be cup product with an ample divisor class. The hard Lefschetz theorem makes Ld−i:Hi(X)→H2d−i(X)L^{d-i}:H^i(X)\to H^{2d-i}(X) an isomorphism for i<di<d, and conjecture B(X)B(X) asserts that its inverse is algebraic, that is, induced by a correspondence in CH⁡i(X×X)⊗Q\operatorname{CH}^i(X\times X)\otimes\mathbb Q. It was posed by Grothendieck as part of his suite of standard conjectures on algebraic cycles [Grothendieck1969Standard], with a systematic early treatment by Kleiman [Kleiman1968Standard].

Conjecture BB is the member of the suite most often isolated, since it implies that the Künneth projectors are algebraic and underpins the intended good behaviour of the category of pure motives; André's book gives a modern account of these implications [Andre2004Motives]. The conjecture is known for curves, surfaces, and abelian varieties, and for various classes of varieties whose motives are generated by such cases.

For arbitrary smooth projective varieties it remains open; a proof would have to exhibit algebraic cycles inverting the Lefschetz isomorphisms, which no general construction currently provides.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.