scholarly journals General Hodge conjecture for abelian varieties of CM-type

2002 ◽  
Vol 78 (6) ◽  
pp. 72-75
Author(s):  
Fumio Hazama
Keyword(s):  
Abelian Varieties ◽  
Hodge Conjecture

ABELIAN VARIETIES OF TYPE III AND THE HODGE CONJECTURE

1999 ◽  
Vol 10 (06) ◽  
pp. 667-675 ◽  
Author(s):  
SALMAN ABDULALI
Keyword(s):  
Abelian Varieties ◽  
Hodge Conjecture ◽  
Type Iii ◽  
General Member

We show that the algebraicity of Weil's Hodge cycles implies the usual Hodge conjecture for a general member of a PEL-family of abelian varieties of type III. We deduce the general Hodge conjecture for certain 6-dimensional abelian varieties of type III, and the usual Hodge and Tate conjectures for certain 4-dimensional abelian varieties of type III.

scholarly journals Notes on Absolute Hodge Classes

2017 ◽  
Author(s):  
Franc¸ois Charles ◽  
Christian Schnell
Keyword(s):  
Abelian Varieties ◽  
Hodge Conjecture ◽  
De Rham Cohomology ◽  
Galois Action ◽  
De Rham ◽  
Definition Of ◽  
Full Proof

This chapter surveys the theory of absolute Hodge classes. First, the chapter recalls the construction of cycle maps in de Rham cohomology, which is then used in the definition of absolute Hodge classes. The chapter then deals with variational properties of absolute Hodge classes. After stating the variational Hodge conjecture, the chapter proves Deligne's principle B and discusses consequences of the algebraicity of Hodge bundles and of the Galois action on relative de Rham cohomology. Finally, the chapter provides some important examples of absolute Hodge classes: a discussion of the Kuga–Satake correspondence as well as a full proof of Deligne's theorem which states that Hodge classes on abelian varieties are absolute.

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