Bloch’s conjecture for complex surfaces

The explanation

Mark three points a, b and c on a flat plane, extended to include its points at infinity. Write the bookkeeping sum 3a − 2b + c. Such a formal sum of points, with whole-number coefficients, is called a zero-cycle. Its degree is the sum of the coefficients, here 3 − 2 + 1 = 2. On this plane, the degree is all that matters. Any two points lie on a line. Along a line, one point can be slid to the other by an allowed move, called rational equivalence. So 3a − 2b + c is equivalent to 2 times any single point.

The rule comes from curves. On a curve inside the surface, a ratio of polynomials has zeros and poles. The zeros minus the poles count as a cycle equivalent to nothing. The Chow group of zero-cycles collects all zero-cycles, up to these movesSource 1. The question is how much of this group survives beyond the degree.

The answer depends on special forms on the surface. A holomorphic two-form is a smooth complex way to measure small areas on the surface. Their count is written p_g. Mumford showed in 1969 that one such form makes the zero-cycles uncontrollably largeSource 2. They then admit no description by finitely many parameters. A second number, q, counts holomorphic one-forms. These give the Albanese variety, a torus that records what a zero-cycle of degree 0 can still detect. The map from zero-cycles of degree 0 to this torus is the Albanese map.

Bloch's conjecture predicts the converse of Mumford's result. If p_g is 0, the degree and the Albanese map should account for every zero-cycleSource 3. When q is also 0, the torus is a single point. Then any two points of the surface should be equivalentSource 4. In 1976, Bloch, Kas and Lieberman proved this for every surface that is not of general typeSource 5. Surfaces of general type are the most twisted class. So the open problem was general-type surfaces with p_g = q = 0Source 6.

These surfaces are rare and hard to build. For a minimal surface of general type with p_g = q = 0, one has b₂ = 10 − K²Source 7. Here b₂ counts independent two-dimensional loops, and K² is a number between 1 and 9 attached to the surface. A Godeaux surface has K² = 1, so b₂ = 9. A surface with K² = 9 has b₂ = 1, like the plane. Earlier proofs needed extra structure on the surface, such as symmetries or quotient picturesSource 8. Voisin handled Catanese and Barlow surfaces in 2014 through families of surfaces. Each method covered only some constructions.

This paper claims to prove Bloch's conjecture for every smooth projective complex surface with p_g = 0Source 9. It treats p_g = q = 0 with no assumption on how the surface was builtSource 10. The remaining surfaces come from Bloch, Kas and LiebermanSource 11. The argument needs no special family, cover or symmetry of the surfaceSource 12. When p_g = q = 0, the paper also claims a consequence for powers of the surface. Every rational class on them is fixed by its cohomology classSource 13.

The obstacle is that loops and forms do not see rational equivalence. For these surfaces, all the loop data is algebraic. Even so, identities in cohomology do not settle rational equivalence by themselvesSource 14. The paper builds a relation among cycles on two copies of the surface. In it, the diagonal appears with a nonzero coefficient. The diagonal acts as the identity on zero-cycles, while the other terms kill zero-cycles of degree 0Source 15. So degree-0 zero-cycles vanish, after allowing fractions. A known torsion theorem then removes the fractions. The relation comes from counting quotients of line bundles on the surface, with weights from a torus symmetry.

Recent preprints by Guletskii and Banerjee also propose proofs with overlapping scopeSource 16. The paper cites them as proposed proofs and uses a different method.

How sure is this? No Lean formalization is listed for this result. The proof rests on long calculations in enumerative geometry that no computer has checked. No specialist review has been published.

What it doesn't show. This does not say anything about surfaces that carry a nonzero holomorphic two-form.