Lefschetz standard conjecture

A complex algebraic shape carries topological data called cohomology. A standard operation moves that data between complementary degrees and is known to be reversible. Can the reverse also be made from algebraic pieces of the shape paired with itself? The conjecture concerns smooth complex projective shapes and rational cohomology; the known reversibility does not itself supply an algebraic construction.[1]

Hard Lefschetz gives a reversible operation on a smooth projective shape’s rational cohomology. The open question asks whether the reverse is itself realized by algebraic pieces in the product shape, rather than only by an abstract inverse linear map. A rational combination of these pieces gives a geometric operation on cohomology called an algebraic correspondence.

For smooth projective $X/\mathbb C$ of dimension $d$ and an ample divisor class $L$, is the inverse of $L^{d-i}:H^i(X,\mathbb Q)\xrightarrow{\sim}H^{2d-i}(X,\mathbb Q)$ induced by a rational algebraic cycle on $X\times X$ for every $0\le i\le d$? The forward isomorphism is known; algebraicity of its inverse is the additional requirement.[1] [2]

Published proofs cover special families, including certain ten-dimensional shapes with additional fibration structure. They do not construct the inverse for every smooth complex projective shape. A later preprint claims a broader OG10 family, still within a special class.[1] [2]

The reported OG10 extension is a preprint result within a hyperkähler deformation class; it does not establish algebraicity of inverse Lefschetz maps in general.[1] [2]

Cohomology compares shapes through topological data, while algebraic pieces provide geometric operations that can be composed and compared. An algebraic reverse would keep this important symmetry inside that geometric language. This matters for the attempt to organize different cohomological descriptions through a common system of algebraic correspondences.[1]

Motives organize cohomological information using algebraic correspondences, whose geometric origin allows operations to be composed within one cycle-based framework. Algebraicity of the Lefschetz inverse would bring a central cohomological symmetry into that framework. Over $\mathbb C$, it implies the other standard conjectures; the Lagrangian-fibration work also investigates when such algebraic control can pass from bases and fibres to the total variety. These are structural consequences, not a proof for arbitrary varieties.[1]