Posted by martin on Tuesday, 26 April 2011 at 12:35
In this post I will define dual abelian varieties over the complex numbers. The motivation is that polarisations can be interpreted as isogenies from an abelian variety to its dual. For the moment, all this is tied to Hodge structures so only works over the complex numbers, but this is the view of polarisations which will we will generalise later to other fields.
Posted by martin on Saturday, 26 March 2011 at 16:07
Last time, we defined polarisations on 
Hodge structures and saw that if 
is a complex abelian variety, then 
has a polarisation.
This time we will prove the converse: if 
is a complex torus such that 
has a polarisation,
then is an abelian variety (in other words, can be embedded in projective space).
The proof is based on studying invertible sheaves on .
This is long, even though I have left out all the messy calculations. For full details, see Mumford’s Abelian Varieties or Birkenhake-Lange’s Complex Abelian Varieties. For the next post, you will only need to know the two statements labelled as theorems.
This theorem is a special case of the Kodaira Embedding Theorem, which tells you that any compact complex manifold is projective if it has a polarisation, but that is somewhat more difficult.
Posted by martin on Saturday, 26 February 2011 at 18:27
In the last post, I discussed Hodge symplectic forms.
Now I shall show that the of an abelian variety has a polarisation, which is defined to be a Hodge symplectic form satisfying a positivity condition.
The importance of polarisations is that they give a way of recognising which Hodge structures come from abelian varieties - I shall discuss this application next time.
Posted by martin on Saturday, 18 December 2010 at 15:00
Both the Hodge structure and the Tate module of an abelian variety come with symplectic forms which are (almost) preserved by the action of the relevant group (Mumford-Tate or Galois group). The form on the Tate module, called the Weil pairing, will require some preparation. So in this post I will construct the Hodge symplectic forms (also called the Riemann forms) on the Hodge structure. Next time I will discuss some further properties of Hodge forms.
Posted by martin on Saturday, 27 November 2010 at 16:22
In this post, I will continue to talk about the 
-adic representations attached to abelian varieties, and in particular the images 
of these representations.
I will define algebraic groups approximating , which are often more convenient to work with.
I will end by stating the Mumford-Tate conjecture, linking to the Mumford-Tate group.
Posted by martin on Sunday, 21 November 2010 at 17:32
I said after my last post that I would write something about -adic representations coming from abelian varieties.
I have finally got around to doing so: here I will tell the story of how these representations are defined, and show that the Tate module is canonically isomorphic to 
.
Next time I will relate this to Mumford-Tate groups.
Posted by martin on Monday, 04 October 2010 at 12:37
In this post I will define the Mumford-Tate group of an abelian variety.
This is a 
-algebraic group, such that the Hodge structure is a representation of this group.
The Mumford-Tate group is important in the study of Hodge theory, and surprisingly also tells us things about the -adic representations attached to the abelian variety.
Posted by martin 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.
Posted by martin on Monday, 10 May 2010 at 21:41
Singular points in a curve are places where curve fails to be smooth: intuitively, multiple points of the curve pile up on top of each other. In this post I will describe a simple invariant of curve singularities, the multiplicity, which essentially counts how many points are piled up there. In the simplest case of an ordinary multiple point, I describe how to use the previous post’s algorithm to compute a power series for each branch of the curve near the singularity.
Posted by martin on Monday, 05 April 2010 at 22:34
An algebraic function is a function which we obtain by solving a polynomial in two variables 
and 
to write as a function of . In general polynomials have more than one root, so (informally) we get a multi-valued function. In this post I will restrict attention to regions of the plane in which we can unambiguously pick a single “branch” of the function. What happens where branches meet will be the subject of a later post.
I shall give an algorithm for expressing an algebraic function (in such a nicely behaved region) as a power series, thereby proving that a power series solution to the original polynomial exists. The generalisation of this result to a complete discrete valuation ring is Hensel’s lemma, and is particularly important to number theorists in the case of p-adic integers (which were invented by Hensel). In this post I will focus on the case of algebraic functions, as it is easier to apply geometric intuition.