Some Surfaces of General Type for Which Bloch’s Conjecture Holds

We give many examples of surfaces of general type with pg = 0 for which Bloch’s conjecture holds, for all values of K2 6= 9. Our surfaces are equipped with an involution. Let S be a smooth complex projective surface with pg(S) = 0. Bloch’s conjecture states that the Albanese map A0(S)0 → Alb(S) is an isomorphism, where A0(S)0 is the Chow group of 0-cycles of degree 0 on S. It is known for all surfaces except those of general type (see [BKL]). For a surface S of general type with pg(S) = 0 we also have q(S) = 0, i.e., Alb(S) = 0 and the canonical divisor satisfies 1 ≤ K ≤ 9. In the decades since this conjecture was formulated, surfaces of general type have become somewhat better understood. Two key developments have been (i) the results of S.Kimura on finite dimensional motives in [Ki] and (ii) the notion of the transcendental motive t2(S) which was introduced in [KMP]. This includes the theorem that if S is a surface with pg(S) = q(S) = 0 then Bloch’s conjecture holds for S iff t2(S) = 0; see Lemma 1.5. In this paper we give motivic proofs of Bloch’s conjecture for several examples of surfaces of general type for each value of K between 1 and 8. This includes some numerical Godeaux surfaces, classical Campedelli surfaces, Keum-Naie surfaces, Burniat surfaces and Inoue’s surfaces. All these surfaces carry an involution, and many were previously known to satisfy Bloch’s conjecture. We can say nothing about the remaining case K=9, because a surface of general type with pg=0 and K=9 has no involution ([DMP, 2.3]). Bloch’s conjecture is satisfied by all surfaces whose minimal models arise as quotients C1×C2/G of the product of two curves of genera ≥2 by the action of a finite group G. A partial classification of these surfaces has been given in [BCGP] and [BCG, 0.1]; the special case where G acts freely only occurs when K S = 8. We also show in Corollary 7.8 that Bloch’s conjecture holds for surfaces with an involution σ for which Date: January 28, 2015.