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Math-27 · mathematics · Algebraic geometry

The Standard Conjectures on Algebraic Cycles

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $X$ be a smooth projective variety over a field, and use a Weil cohomology theory with its cycle-class map. The standard conjectures ask that the cohomological operators and relations dictated by algebraic cycles actually have algebraic realizations. In particular, are the inverse hard-Lefschetz operators induced by algebraic correspondences, and are the Künneth projectors onto the individual cohomological degrees algebraic? The collection also includes the agreement of numerical and homological equivalence and the positivity assertion for primitive algebraic classes.

Formulation references: [S1], [S2].

Short English statement

Let $X$ be a smooth projective variety over a field, and use a Weil cohomology theory with its cycle-class map. The standard conjectures ask that the cohomological operators and relations dictated by algebraic cycles actually have algebraic realizations. In particular, are the inverse hard-Lefschetz operators induced by algebraic correspondences, and are the Künneth projectors onto the individual cohomological degrees algebraic? The collection also includes the agreement of numerical and homological equivalence and the positivity assertion for primitive algebraic classes.

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Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-27-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-27

Manuscript 1 (family 032, 2026-09-30): The explicit Hodge-standard corollary gives positivity and numerical/etale homological equivalence for abelian varieties in arbitrary characteristic, while the Standard Conjectures record covers all smooth projective varieties and its remaining conjectures.

The Standard Conjectures on Algebraic Cycles

OpenAI's claim

The manuscripts claim algebraicity of rational Hodge classes on CM abelian varieties, with consequences for generalized Hodge classes and finite-field Tate classes. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
A complex abelian variety $A$ is CM if $\operatorname{End}(A)\otimes\mathbb Q$ contains a commutative semisimple algebra of dimension $2\dim A$. For every such $A$ and every $p\ge0$, OpenAI claims \[ \operatorname{CH}^p(A)_{\mathbb Q}\longrightarrow H^{2p}(A,\mathbb Q)\cap H^{p,p}(A) \quad\text{is surjective}. \] This also applies to arbitrary finite products of CM abelian varieties, including their powers. If $W\subset H^k(A,\mathbb Q)$ is a rational Hodge substructure and $W(r)$ is effective (all Hodge bidegrees are nonnegative), it claims a closed algebraic $Z\subset A$ of codimension at least $r$ with $W\subset\ker[H^k(A,\mathbb Q)\to H^k(A\setminus Z,\mathbb Q)]$. The manuscript additionally asserts that, for every abelian variety $A/\mathbb F_q$, every $r\ge0$ and prime $\ell\ne\operatorname{char}\mathbb F_q$, \[ \operatorname{CH}^r(A)\otimes\mathbb Q_\ell\longrightarrow H^{2r}_{\mathrm{\acute et}}(A_{\overline{\mathbb F}_q},\mathbb Q_\ell(r))^{\operatorname{Gal}(\overline{\mathbb F}_q/\mathbb F_q)} \quad\text{is surjective}. \] Numerical and $\ell$-adic homological equivalence are claimed to agree on rational cycles. Over any algebraically closed field, for an abelian $g$-fold and rational ample class $h$, the form $(-1)^r\deg(xyh^{g-2r})$ is claimed positive definite on the real extension of $\ker[h^{g-2r+1}:A^r_{\mathrm{num}}\to A^{g-r+1}_{\mathrm{num}}]$ for $0\le r\le g/2$.

Outline of the argument

Manuscript 1 (family 032, 2026-09-30): The argument realizes four-factor balanced tensors using unitary theta constructions. Elementary switches between equal-sum lists, followed by algebraic polarization contractions and descent of coefficients, build arbitrary balanced CM tensors from these elementary cycle classes.

Formal verification

Family 032 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

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