Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- List of Conference Participants
- Frontispiece
- PART I HODGE THEORY AT THE BOUNDARY
- PART II ALGEBRAIC CYCLES AND NORMAL FUNCTIONS
- 7 A simple construction of regulator indecomposable higher Chow cycles in elliptic surfaces
- 8 A relative version of the Beilinson-Hodge conjecture
- 9 Normal functions and spread of zero locus
- 10 Fields of definition of Hodge loci
- 11 Tate twists of Hodge structures arising from abelian varieties
- 12 Some surfaces of general type for which Bloch's conjecture holds
- PART III THE ARITHMETIC OF PERIODS
12 - Some surfaces of general type for which Bloch's conjecture holds
from PART II - ALGEBRAIC CYCLES AND NORMAL FUNCTIONS
Published online by Cambridge University Press: 05 February 2016
- Frontmatter
- Contents
- Preface
- Introduction
- List of Conference Participants
- Frontispiece
- PART I HODGE THEORY AT THE BOUNDARY
- PART II ALGEBRAIC CYCLES AND NORMAL FUNCTIONS
- 7 A simple construction of regulator indecomposable higher Chow cycles in elliptic surfaces
- 8 A relative version of the Beilinson-Hodge conjecture
- 9 Normal functions and spread of zero locus
- 10 Fields of definition of Hodge loci
- 11 Tate twists of Hodge structures arising from abelian varieties
- 12 Some surfaces of general type for which Bloch's conjecture holds
- PART III THE ARITHMETIC OF PERIODS
Summary
ABSTRACT. We give many examples of surfaces of general type with pg = 0 for which Bloch's conjecture holds, for all values of K2 ≠ 9. Our surfaces are equipped with an involution.
Let S be a smooth complex projective surface with pg(S) = 0. Bloch'sconjecture 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 ≤ K2 ≤ 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'sconjecture 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 K2 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'sconjecture. We can say nothing about the remaining case K2 =9, because a surface of general type with pg=0 and K2=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 K2S = 8. We also show in Corollary 7.8 that Bloch'sconjecture holds for surfaces with an involution σ for which K2 = 8 and S/σ is rational.
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- Recent Advances in Hodge TheoryPeriod Domains, Algebraic Cycles, and Arithmetic, pp. 308 - 330Publisher: Cambridge University PressPrint publication year: 2016
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