Bloch's conjecture for complex surfaces at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Bloch's conjecture for complex surfaces

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	shapes made of equations	Levels:	1
Category:	Algebraic and complex geometry	Lean version:	not yet
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Bloch’s conjecture for complex surfaces. 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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LEVEL 1 Bloch’s conjecture for surfaces with p_g=0

released 2026-09-24 | 2 theorems · 20 lemmas · 33 proofs · 20,344 words | PLAY LEVEL 1 » (pdf)

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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