On the Hard Lefschetz Conjecture for Lawson Homology

Abstract

Friedlander and Mazur proposed a conjecture of hard Lefschetz type for Lawson homology. We shall relate this conjecture to Suslin's conjecture for Lawson homology and hence to Grothendieck's standard conjectures. Applying the SL ₂-action on abelian varieties observed by Beauville, we show that for abelian varieties, this conjecture is equivalent to a weak form of a vanishing conjecture of Beauville type for Lawson homology.

Key Words:

• Abelian variety
• Algebraic cycle
• Lawson homology

2010 Mathematics Subject Classification:

• 14F43
• 14C25
• 11G10

ACKNOWLEDGEMENTS

The author would like to thank Professor Baohua Fu for encouraging him to write this note, and for valuable discussions and suggestions. The author thanks also Professor Eric Friedlander and Professor Wenchuan Hu for valuable suggestions and comments. Finally, the author would like to thank the Referee for pointing out some defects and for giving useful suggestions.

The author is partially supported by National Natural Foundation of China (10871106).

Notes

Communicated by C. Pedrini.