Hodge structures and abelian varieties
Posted by Martin Orr on Friday, 24 September 2010 at 08:48
I spend most of my time thinking about the Hodge structures attached to abelian varieties, so I decided that I should explain what these Hodge structures are. A Hodge structure is a type of algebraic structure found on the (co)homology of complex projective varieties.
Here I will discuss only the special case of the first homology of abelian varieties. This is the simplest case, but is nonetheless very important. In particular, the Hodge structures on other homology and cohomology groups for abelian varieties can be calculated from that of the first homology. Also Hodge structures on the first (but not higher) cohomology of non-abelian varieties can be reduced to the case of abelian varieties by passing to something called the Albanese variety, generalising the Jacobian of curves.