Result 040, Algebraic and complex geometry

Bloch’s conjecture for complex surfaces

Proves Bloch's conjecture: for every smooth connected projective complex surface S with pg(S)=0p_g(S)=0, the Albanese map CH0(S)0→Alb(S)(C)\mathrm{CH}_0(S)^0\to\mathrm{Alb}(S)(\mathbb C) on integral degree-zero zero-cycles is an isomorphism. This combines the new pg=q=0p_g=q=0 theorem with the classical theorem of Bloch, Kas, and Lieberman.

Proof

The bigger picture

Why it matters

Integer-weighted collections of points on an algebraic surface can hide subtle geometric information. The manuscript claims that, for surfaces with no holomorphic two-forms, a geometric torus completely determines these collections up to the relevant equivalence.

What changes?

The manuscript reports Bloch's conjecture for every smooth, connected, projective complex surface with geometric genus p_g equal to zero, meaning it has no nonzero holomorphic two-forms. A degree-zero zero-cycle is a finite integer-weighted sum of points whose weights sum to zero. The claim is that the Albanese map identifies these cycles, modulo rational equivalence, exactly with the complex points of the Albanese variety, a torus encoding holomorphic one-forms. The statement retains integer coefficients, rather than allowing rational coefficients.

What does that help mathematicians do?

Consequently, two degree-zero cycles are rationally equivalent exactly when they have the same Albanese image. Rational equivalence identifies point combinations through divisors of rational functions on curves in the surface. If the Albanese variety is trivial, any two cycles of the same degree are equivalent. This rules out additional zero-cycle information invisible to the torus, giving researchers a complete criterion for equivalence under the stated assumptions.

Are there practical applications?

The immediate value is foundational: the claimed theorem replaces an abstract group of point combinations and relations with a geometric object that records them completely. It provides a uniform description of zero-cycles across the stated class of surfaces. The supplied material describes no practical deployment; its relevance is to understanding algebraic cycles and the geometry of complex surfaces.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Bloch’s conjecture for surfaces with p_g=0

September 24, 2026 45 pages

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

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Cite (BibTeX) 
@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}
}

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