Math Conjectures

Bloch Conjecture for Surfaces with pg=0p_g=0

OPENMajorConjectureProposed 1975 · Full conjecture

Canonical statement

Let XX be a smooth projective complex surface with pg(X)=h0(X,KX)=0p_g(X)=h^0(X,K_X)=0. The Albanese map
alb⁡X:CH⁡0(X)0⟶Alb⁡(X)(C) \operatorname{alb}_X: \operatorname{CH}_0(X)^0\longrightarrow \operatorname{Alb}(X)(\mathbb C)
from degree-zero zero-cycles modulo rational equivalence is an isomorphism.
View source LaTeX
Let \(X\) be a smooth projective complex surface with \(p_g(X)=h^0(X,K_X)=0\). The Albanese map
\[
  \operatorname{alb}_X:
    \operatorname{CH}_0(X)^0\longrightarrow
    \operatorname{Alb}(X)(\mathbb C)
\] from degree-zero zero-cycles modulo rational equivalence is an isomorphism.
AG-ZERO-020:[Bloch1975K2][BlochKasLieberman1976][Voisin2014Bloch]
Algebraic geometryAlgebra and representation theory

Notes

Bloch conjectured in the mid-1970s that for a smooth projective complex surface XX with geometric genus pg(X)=0p_g(X)=0, the Chow group of zero-cycles is as small as cohomology allows: the Albanese map CH⁡0(X)0→Alb⁡(X)\operatorname{CH}_0(X)^0\to\operatorname{Alb}(X) on degree-zero cycles modulo rational equivalence should be an isomorphism. The conjecture emerged from Bloch's study of K2K_2 and algebraic cycles [Bloch1975K2], and it is the converse companion to Mumford's classical theorem that pg>0p_g>0 forces CH⁡0\operatorname{CH}_0 to be infinite-dimensional.

Bloch, Kas and Lieberman established the conjecture for all surfaces with pg=0p_g=0 that are not of general type, using the classification of surfaces [BlochKasLieberman1976]. For surfaces of general type with pg=0p_g=0 the conjecture has been verified only case by case; notably, Voisin proved it for Catanese surfaces and Barlow surfaces [Voisin2014Bloch].

No argument covering all general-type surfaces of geometric genus zero is known; a resolution requires either a general mechanism forcing the Albanese map to be an isomorphism whenever pg=0p_g=0, or a counterexample among general-type surfaces. In that generality the conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.