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Math-2324 · mathematics · Algebraic geometry

Bloch–Beilinson Torsion Conjecture for Albanese Kernels

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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.
For a smooth projective variety $V$ over a number field $k$, is the kernel of its Albanese map $\varphi_V$ torsion?

Formulation references: [S1], [S2].

Short English statement

For a smooth projective variety $V$ over a number field $k$, is the kernel of its Albanese map $\varphi_V$ torsion?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2324-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2324

Manuscript 1 (family 040, 2026-09-24): For a number-field surface in the stated p_g=0 class, the complex integral Albanese statement kills the kernel after base change; descent of a rational equivalence over a finite extension and the norm give a torsion kernel over the number field. The problem quantifies over all smooth projective varieties and dimensions, which the paper does not cover.

Bloch–Beilinson Torsion Conjecture for Albanese Kernels

OpenAI's claim

For every smooth complex projective surface with $p_g=0$, OpenAI claims that the Albanese map on integral degree-zero cycles is an isomorphism. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For a smooth connected projective complex surface $S$ put $p_g=h^0(S,\omega_S)$ and $q=h^1(S,\mathcal O_S)$. OpenAI asserts the integral statements \[ p_g=q=0\ \Longrightarrow\ \deg:\operatorname{CH}_0(S)\xrightarrow{\sim}\mathbb Z, \qquad p_g=0\ \Longrightarrow\ \operatorname{alb}_S:\operatorname{CH}_0(S)^0\xrightarrow{\sim}\operatorname{Alb}(S)(\mathbb C), \] where $\operatorname{CH}_0(S)^0$ is the subgroup of degree-zero zero-cycles modulo rational equivalence. Under $p_g=0$, it also claims that the rational Chow motive is finite-dimensional in Kimura's sense and has zero transcendental degree-two summand. When $p_g=q=0$ it claims \[ h(S)_{\mathbb Q}\simeq\mathbf1\oplus\mathbb L^{\oplus b_2(S)}\oplus\mathbb L^2, \] where $\mathbb L$ is the Lefschetz motive. Under the joint hypothesis $p_g=q=0$, for each power $Y=S^m$ and each Hilbert scheme $Y=S^{[n]}$, the corresponding rational motive is pure Tate and $\operatorname{CH}^i(Y)_{\mathbb Q}\xrightarrow{\sim}H^{2i}(Y,\mathbb Q)$, with $H^{2i+1}(Y,\mathbb Q)=0$, for every $i\ge0$.

Outline of the argument

Manuscript 1 (family 040, 2026-09-24): A determinant-constrained Quot obstruction theory and torus localization produce a Chow relation involving the diagonal. A nested-Hilbert recursion leaves two surface factors unintegrated, retaining an actual correspondence identity; its external-product terms annihilate degree-zero cycles, and the diagonal coefficient gives the required vanishing after the standard reductions.

Formal verification

Family 040 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.

Sources

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