OpenAI claims Bloch’s conjecture for surfaces

Zero-cycles on a surface are formal sums of points, and two cycles are rationally equivalent when they differ by divisors of rational functions on curves. Mumford showed in 1969 that a surface with a nonzero holomorphic 2-form has an enormous, infinite-dimensional group of zero-cycles. Spencer Bloch conjectured the converse: if p_g = 0, then a zero-cycle is determined by its degree and its image in the Albanese variety. Bloch, Kas and Lieberman proved this for surfaces that are not of general type, leaving the general-type surfaces with p_g = q = 0, for which the prediction is that any two points are rationally equivalent. Since then the conjecture has been checked only for particular families such as Godeaux, Campedelli, Barlow and Catanese surfaces.

OpenAI’s 45-page manuscript claims the general-type case with no assumptions on how the surface is built. Instead of using symmetries or explicit families, it produces the needed relation among zero-cycles by virtual localization on a Quot scheme of the surface, using derived moduli and enumerative-geometry techniques. It also states consequences for Chow motives of powers and Hilbert schemes of S.

Bloch’s conjecture is one of the central open problems about algebraic cycles, a test case of the Bloch–Beilinson conjectures. A uniform proof would replace fifty years of case-by-case work, but at present it rests on an unrefereed, unformalized manuscript.

Daniel Litt, who had worked on the problem as a graduate student, welcomed the claim but noted that the paper cites some earlier work he considers wrong without saying so, adding that it does not seem to rely on it. The manuscript itself lists a 2025 proof proposal by Guletskiĭ covering the same statement as “proposed”. No specialist has published a reading of the argument.