Algebraic and complex geometry · Result 040

Bloch’s conjecture for complex surfaces

1 paper

Proves Bloch's conjecture: for every smooth connected projective complex surface S with \(p_g(S)=0\), the Albanese map \(\mathrm{CH}_0(S)^0\to\mathrm{Alb}(S)(\mathbb C)\) on integral degree-zero zero-cycles is an isomorphism. This combines the new \(p_g=q=0\) theorem with the classical theorem of Bloch, Kas, and Lieberman.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 1

Bloch’s conjecture for surfaces with p_g=0

OpenAI · September 24, 2026

We prove Bloch's conjecture for smooth connected projective complex surfaces with \(p_g=0\): the Albanese homomorphism on integral degree-zero zero-cycles is an isomorphism.

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Cite

@misc{OAI:Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026,
  author = {{OpenAI}},
  title = {{Bloch's conjecture for surfaces with $p_g=0$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf}{OAI:Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.