Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

The Hodge- ${\mathcal D}$-conjecture for $\text{\rm K3}$ and Abelian surfaces


Authors: Xi Chen and James D. Lewis
Journal: J. Algebraic Geom. 14 (2005), 213-240
DOI: https://doi.org/10.1090/S1056-3911-04-00390-X
Published electronically: December 30, 2004
MathSciNet review: 2123228
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Abstract | References | Additional Information

Abstract: Let $X$ be a projective algebraic manifold, and $\text{CH}^{k}(X,1)$ the higher Chow group, with corresponding real regulator $\text{r}_{k,1}\otimes {{\mathbb R}}: \text{CH}^k(X, 1)\otimes {{\mathbb R}} \to H_{\mathcal D}^{2k-1}(X,{{\mathbb R}}(k))$. If $X$ is a general K3 surface or Abelian surface, and $k=2$, we prove the Hodge- ${\mathcal D}$-conjecture, i.e. the surjectivity of $\text{r}_{2,1}\otimes {{\mathbb R}}$. Since the Hodge- ${\mathcal D}$-conjecture is not true for general surfaces in ${\mathbb P}^{3}$ of degree $\geq 5$, the results in this paper provide an effective bound for when this conjecture is true.


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Additional Information

Xi Chen
Affiliation: 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: xichen@math.ualberta.ca

James D. Lewis
Affiliation: 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA
Email: lewisjd@gpu.srv.ualberta.ca

DOI: https://doi.org/10.1090/S1056-3911-04-00390-X
Received by editor(s): April 11, 2003
Received by editor(s) in revised form: November 2, 2003
Published electronically: December 30, 2004
Additional Notes: Both authors were partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada

American Mathematical Society