Open problem, Geometry, Algebra, posed 1969

Grothendieck's standard conjectures on algebraic cycles

Open

Let $X$ be a smooth projective variety of dimension $d$ over a field $k$ and $H^*(X)$ a Weil cohomology theory. Conjecture B (Lefschetz type) asserts that the inverse Lefschetz operators are induced by algebraic correspondences on $X \times X$ ; Conjecture D asserts that numerical equivalence of algebraic cycles coincides with homological equivalence with respect to $H^*$ ; Conjecture I (Hodge type) asserts that the Hodge–Riemann bilinear relations hold for the intersection pairing on primitive algebraic cohomology classes.

Research frontier as of 2026

As of 2026, all the standard conjectures remain open for general smooth projective varieties. Progress is confined to special classes: abelian varieties (Lieberman, Kleiman 1968), Grassmannians and flag varieties, smooth complete intersections, uniruled threefolds and unirational fourfolds (Arapura), and hyperkähler varieties of $K3^{[n]}$ -type (Charles, Markman 2013). Conjecture C (algebraicity of Künneth projectors) has additional traction over finite fields via crystalline methods (Katz–Messing).

Best known results

The Lefschetz standard conjecture (B) is proven for abelian varieties (Lieberman, Kleiman 1968), Grassmannians and generalized flag varieties, smooth complete intersections in projective space, and hyperkähler varieties of $K3^{[n]}$ -type (Charles, Markman 2013).
The Hodge standard conjecture (I) is a theorem in characteristic zero via classical Hodge theory but is largely open in positive characteristic beyond the cases covered by Conjecture B.

Tools and where they stop

Tool	Achieved	Where it stops
Explicit algebraic correspondences via theta divisors and Fourier–Mukai transforms	Constructs the inverse Lefschetz operator algebraically for abelian varieties and, via derived-category methods, for hyperkähler varieties of $K3^{[n]}$ -type	Relies on special geometric structure (group law, hyperkähler monodromy) unavailable for a general smooth projective variety
$\ell$ -adic and crystalline cohomology with Lefschetz pencils	Proved the Weil conjectures (Deligne 1974) and Conjecture C over finite fields (Katz–Messing) without needing the full standard conjectures	Does not establish algebraicity of the cycles involved, only their cohomological/numerical properties, so Conjectures B and D remain untouched

Open questions

Does the Lefschetz standard conjecture (B) hold for every smooth projective variety over a field of positive characteristic?
Does numerical equivalence coincide with homological equivalence (Conjecture D) for all smooth projective varieties, at least conjecturally implied by the Standard Conjecture B together with the Hodge or Tate conjecture?

References

Alexander Grothendieck (1969). Standard conjectures on algebraic cycles
David I. Lieberman (1968). Numerical and homological equivalence of algebraic cycles on Hodge manifolds · DOI:10.2307/2373533
François Charles, Eyal Markman (2013). The Standard Conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces · DOI:10.1112/S0010437X12000607 · arXiv:1009.0413