We define an abelian varieties. We define isogenies of abelian varieties and give some properties of isogenies. We prove some basic properties including the following: abelian varieties are commutative, the relative dualizing sheaf is trivial and we prove that all rational maps from a nonsingular variety to an abelian variety extend to a morphism.​

We show that all abelian varieties are projective, not just proper. The proof can be thought of as a generalization of the fact that every elliptic curve embeds into a 2-dimensional projective space via a Weierstrass equation.

In the notes and time permitting, we discuss which n it is possible to embed an abelian variety of dimension g into projective space of dimension n. It turns out that in general, for an abelian variety of dimension g, the smallest such value of n is 2g+1. Note that the case of elliptic curves, all of which can be embedded in 2-dimensional projective space via a Weierstrass equation are an exception.

We define the dual abelian variety of a given abelian variety, explain its universal property and sketch the proof of existence. 

Precise topics to be determined.

Precise topics to be determined.

Precise topics to be determined.

Precise topics to be determined.

Precise topics to be determined.

Precise topics to be determined.

Precise topics to be determined.

Precise topics to be determined.

Precise topics to be determined