Bloch’s conjecture for complex surfaces

1 · The essentials

The manuscript claims a result about smooth complex surfaces that are connected and projective, meaning they sit inside complex projective space. It concerns surfaces satisfying the numerical condition $p_{g} = 0$ . The question is whether certain collections of points on such a surface can be completely described by a standard map to another geometric object. The collections carry whole-number weights whose total is zero, and are compared using an algebraic equivalence rule called rational equivalence. The claimed answer is yes: the map captures every possible output and loses no information about these collections after that comparison. The repository describes this as a resolution of Bloch’s conjecture for these surfaces. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine weighted collections of points as account entries whose credits and debits balance to zero. The claimed result says that a particular summary record captures exactly the information left after the permitted bookkeeping changes; the image does not explain what those mathematical changes are.

Why it matters

The claim identifies the group of integral degree-zero zero-cycles, modulo rational equivalence, with the complex points of the surface’s Albanese variety. According to the repository summary, combining a new result for $p_{g} = q = 0$ with a classical theorem of Bloch, Kas, and Lieberman yields the full statement for $p_{g} = 0$ .

2 · Going further

Let $S$ be a smooth connected projective complex surface with $p_{g} (S) = 0$ . The precise claim is that the Albanese homomorphism $CH_{0} (S)^{0} \to Alb (S) (C)$ is an isomorphism. Here the source is the group of integral degree-zero zero-cycles modulo rational equivalence, and the target is the group of complex points of the Albanese variety. Being an isomorphism means that the map is both one-to-one and onto, and respects the group operations.

The repository summary describes the result as a combination of a new theorem for the case $p_{g} = q = 0$ and the classical theorem of Bloch, Kas, and Lieberman. This gives the stated structure of the argument, but neither the summary nor the supplied abstract explains the mechanism of the new theorem or the details of the combination. Only one manuscript and its abstract are supplied, so no relationship to companion manuscripts can be described from this material. The abstract states the full $p_{g} = 0$ claim; it does not provide an argument that can be assessed here.

Useful notions

Smooth projective complex surfaces
Zero-cycles and their degree
Rational equivalence
Groups and isomorphisms
The Albanese map

What was known before / what is new

Before

The repository summary cites a classical theorem of Bloch, Kas, and Lieberman as an ingredient. The supplied text does not state that theorem separately or describe its precise hypotheses.

New

The repository identifies the $p_{g} = q = 0$ theorem as new. It claims that combining this theorem with the cited classical result establishes the Albanese isomorphism for every smooth connected projective complex surface with $p_{g} = 0$ .

Limits and assumptions

The statement requires a smooth, connected, projective surface over the complex numbers with $p_{g} = 0$ , and concerns integral degree-zero zero-cycles. No conclusion for surfaces outside these hypotheses is supplied. The abstract contains no details of the argument, and the main result is not formalized in Lean and has not been formally verified. The supplied material does not identify any remaining open cases or questions.

Mini glossary

Integral zero-cycle: A finite collection of points with integer weights, written as a formal sum.
Degree-zero zero-cycle: In this complex-surface setting, a zero-cycle whose integer weights sum to zero.
Rational equivalence: The algebraic rule for identifying zero-cycles when forming the group $CH_{0} (S)$ .
Albanese map: The homomorphism from degree-zero zero-cycle classes to the complex points of the Albanese variety that appears in the conjecture.
Isomorphism: A one-to-one and onto map that preserves the mathematical structure, here the group operations.

Keywords : Bloch’s conjecture · Zero-cycles · Albanese map

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves Bloch's conjecture: for every smooth connected projective complex surface S with $p_{g} (S) = 0$ , the Albanese map $CH_{0} (S)^{0} \to Alb (S) (C)$ 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.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

★

Bloch’s conjecture for surfaces with p_g=0

24/09/2026

To be verified

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.

PDF GitHub folder