Analytic Theory of Abelian Varieties I

It will be useful to have past posts as reference: one-parameter subgroups, exponential map, exponential properties, and lie algebra actions.

As a summary and way to get notation going I’ll just list some important things that are proved there and that we’ll use. First off, everything in those posts used smooth manifolds, but we’ll be using complex manifolds. This just means transition maps must be holomorphic rather than smooth. We can still do all of those things with \mathbb{C} in place of \mathbb{R} just making sure that everything is actually holomorphic rather than only smooth.

Let X be a connected complex manifold of dimension g with a group structure such that inversion X\to X by x\mapsto x^{-1} and multiplication X\times X\to X by (x,y)\mapsto xy is holomorphic. This is called a connected complex Lie group.

Let V=T_eX be the tangent space at the identity. This is a complex vector space of dimension g. If v\in V, then we get an entire left-invariant vector field on X by defining Y_x=(dL_x)_e(v) where L_x is left multiplication by x. Any left-invariant vector field turns out to automatically be holomorphic. The set of all left-invariant vector fields on X is denoted \mathcal{L}(X) and it called the Lie algebra associated to the Lie group X.

Thus we get an integral curve of the flow of this vector field associated to v. Since the vector field is complete, we get the one-parameter subgroup \phi_v: \mathbb{C}\to X. See the post on this for more rigor. This map satisfies d\phi_v(\frac{d}{dz})=v.

Note that we can always identify \mathcal{L}(\mathbb{C}) with \mathbb{C} by the isomorphism w\mapsto w\frac{d}{dz}. Under this identification, we have d\phi_v(1)=v. The map \phi_v(t): \mathbb{C}\times V\to X is holomorphic. Define exp: V\simeq \mathcal{L}(X)\to X by exp(v)=\phi_v(1).

See the exponential map post to see why we get some nice properties such as \phi_{sv}(t)=\phi_v(st). If we identify the tangent space to V at 0 with V itself, then we get that (dexp)_0(v)=v. Lastly, given any homomorphism of complex Lie groups T: X_1\to X_2 we get that T(exp_{X_1}y)=exp_{X_2}((dT)_ey).

I should probably be explaining the subtleties going on between considering the tangent space as complex versus real. Basically, if you write down all the maps and identifications carefully, all of these things actually respect the \mathbb{R}-structure. But we won’t really worry about that unless it comes up later.

We’ll do one theorem: If X is a compact connected complex Lie group, then X is abelian.

Consider the conjugation map C_x: X\to X by C_x(y)=xyx^{-1}. Then (dC_x)_e: V\to V is an automorphism. We also have that x\mapsto (dC_x)_e a holomorphic map \psi: X\to Aut(V) which is a subspace of End(V). Since X is compact and \psi holomorphic, it must be constant. i.e. (dC_x)_e=(dC_e)_e=id_V.

Since C_x is a homomorphism of complex Lie groups X\to X, we get the exponential property mentioned above: C_x(exp y)=exp((dC_x)_ey)=exp(id_V(y))=exp(y). This tells us that exp(V)=\{exp(v) : v\in V\} is in the center of X. But (dexp)_0 is the identity and in particular has full rank, so by the Implicit Function Theorem exp is a homeomorphism from a neighborhood of 0 in V to a neighborhood of the identity in X.

But X is connected, so any neighborhood of the identity generates all of X. Thus exp(V) generates the whole group X, and hence the center of X is all of X meaning X is abelian.

7 thoughts on “Analytic Theory of Abelian Varieties I

  1. There is the large reference by Birkenhanke (“Complex Abelian Varieties”) which proves the basic facts about Lie groups (in particular, tori), if I remember correctly, almost immediately. (In fact the structure theory of Lie groups is treated beautifully in the book of Brocker and tom Dieck.) An elementary treatment can be found in the book by Murty. In a similar vein, Swinnerton-Dyer has his short survey (which I have admittedly not at all perused). Cornell and Silverman also has an introduction, though James Milne now has a large set of notes on the subject.

    Finally, Shimura has his advanced treatise.

  2. You should be more careful with the citations. The book “by Birkenhanke” has two authors, Birkenhake and Lange. They also wrote, together, a book on Complex Tori more generally.

  3. I’m working through Mumford’s Abelian Varieites. The stuff in this post is essentially assumed on the first page, so it might take another post or two before I really get into it.

    The only book in the library that I found on complex Lie groups to help with this background material was Dong Hoon Lee’s The Structure of Complex Lie Groups. So that is where just standard complex Lie group stuff is coming from. The abelian proof is the first theorem in Mumford.

    Maybe I’ll check out some of the books in pmoduli’s comments.

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