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March 13, 2022

Line Bundles on Complex Tori (Part 1)

Posted by John Baez

A complex abelian variety is a group in the category of smooth complex projective varieties. They’re called that because — wonderfully — they turn out to all be abelian! I’ve been studying holomorphic line bundles on complex abelian varieties, which is a really nice topic with fascinating connections to quantum physics, Jordan algebras and number theory. This is the book that’s helped me the most so far:

  • Christina Birkenhake and Herbert Lange, Complex Abelian Varieties, Springer, Berlin, 2004.

But the subject is so rich that it can be hard to see the forest for the trees! So for my own benefit I’d like to describe the classification of holomorphic line bundles on an abelian variety — or more generally, any ‘complex torus’.

A complex torus is the same as the quotient of a finite-dimensional complex vector space by a lattice. Every abelian variety is a complex torus, but not every complex torus is an abelian variety: you can’t make them all into projective varieties.

I will avoid saying a lot of things people usually say about this subject, in order to keep things short.

Complex line bundles on a topological space XX are classified up to isomorphism by elements of H 2(X,)H^2(X,\mathbb{Z}).

This becomes particularly nice when XX is a real torus. Let’s equip XX with the structure of an abelian Lie group. Then the universal cover X˜\tilde{X} is a finite-dimensional real vector space, and the kernel of the projection

p:X˜X p \colon \tilde{X} \to X

is a lattice

L=kerp L = \ker p

in the vector space X˜\tilde{X} — that is, a free abelian group of rank equal to the dimension of that vector space. This lets us write

X=X˜/L X = \tilde{X}/L

In this situation H 2(X,)H^2(X,\mathbb{Z}) is isomorphic to the group of alternating bilinear maps

A:L×L A \colon L \times L \to \mathbb{Z}

There’s nothing magic about the number 22 and the word ‘bilinear’ here — the same kind of description works for H i(X,)H^i(X, \mathbb{Z}) for any ii, using alternating multilinear maps.

So, complex line bundles on a real torus XX are classified up to isomorphism by alternating bilinear maps A:L×LA \colon L \times L \to \mathbb{Z}. And I should be clear: we could use isomorphism of topological line bundles, or of smooth line bundles: it makes no difference here.

But now suppose X˜\tilde{X} is equipped with the structure of a complex vector space. This makes X=X˜/LX = \tilde{X}/L into a complex manifold, and indeed a compact abelian complex Lie group. We call XX a complex torus. In fact, every compact connected complex Lie group is a complex torus.

Complex tori are a lot subtler than real tori. All real tori of a given dimension are isomorphic. This is not true for complex tori! For example, complex tori of complex dimension 1 — hence real dimension 2, so they look like doughnuts — are called elliptic curves. There’s an interesting space of different isomorphism classes of elliptic curves, called the moduli space of elliptic curves. The story gets even more complicated in higher dimensions.

Now for the real point of this article. If XX is a complex torus, it’s an interesting question to classify holomorphic complex line bundles on XX. How does it work?

This question breaks up nicely into two parts:

Question 1. Which alternating bilinear A:L×LA \colon L \times L \to \mathbb{Z} come from holomorphic complex line bundles?

Question 2. If A:L×LA \colon L \times L \to \mathbb{Z} comes from a holomorphic complex line bundle, how many different such bundles give this particular AA?

A nice thing is that the answer to Question 2 turns out not to depend on the choice of AA, as long as there’s any holomorphic line bundle at all giving that AA.

Answer to Question 1. The answer to this question is simple, though it takes work to show it’s correct. Using the fact that every vector in X˜\tilde{X} is a real linear combination of vectors in the lattice LX˜L \subseteq \tilde{X}, you can show any alternating bilinear map

A:L×LA \colon L \times L \to \mathbb{Z}

extends uniquely to an alternating real-bilinear map

A˜:X˜×X˜ \tilde{A} \colon \tilde{X} \times \tilde{X} \to \mathbb{R}

It then turns out that AA comes from a holomorphic complex line bundle if and only if

A˜(iv,iw)=A˜(v,w) \tilde{A}(i v, i w) = \tilde{A}(v,w)

for all v,wX˜v,w \in \tilde{X}.

Answer to Question 2. This answer is also simple. Suppose AA comes from some holomorphic complex line bundle. Then the set of isomorphism classes of holomorphic complex line bundles giving this AA is X *X^*, the ‘dual’ of torus XX.

What’s dual of a torus? In this game it’s another torus — not the usual Pontryagin dual of the torus. Remember, for any real torus XX we can write

X=X˜/L X = \tilde{X}/L

The real vector space X˜\tilde{X} has a dual X˜ *\tilde{X}^*, defined in the usual way. Sitting inside this dual vector space we have the so-called dual lattice of the lattice LL, defined like this:

L *={fX˜ *:f(v)forallvL} L^\ast = \{ f \in \tilde{X}^\ast : \; f(v) \in \mathbb{Z} \; for \; all \; v \in L \}

So, the quotient X˜ */L *{\tilde{X}}^\ast / L^\ast is a torus, called the dual torus of the torus XX and again denoted with a star:

X *=X˜ */L * X^\ast = {\tilde{X}}^\ast / L^\ast

Now if XX is a complex torus, the vector space X˜\tilde{X} is a complex vector space, and thus so is its dual X˜ *{\tilde{X}}^\ast. Beware: X˜ *{\tilde{X}}^\ast is still defined in the usual way: it’s the real dual of the underlying real vector space of X˜\tilde{X}. But it gets the structure of a complex vector space from that of X˜\tilde{X} — this is a bit tricky, and I’ll talk about it next time. So the dual torus X *X^\ast is a complex torus if XX is.

People like to package up the answers to Question 1 and Question 2 into a single theorem, the Appell–Humbert theorem. And the usual statement of this theorem involves lots of extra jargon — which you have to learn if you want to study this subject:

  • The set of isomorphism classes of holomorphic complex line bundles on a complex torus XX is called its Picard group, Pic(X)\mathrm{Pic}(X). This is actually an abelian group, because you can tensor line bundles. In fact, it’s a complex Lie group.

  • The identity component Jac(X)\mathrm{Jac}(X) of the Picard group is called the Jacobian of XX, or, as if there weren’t enough jargon already, the Picard variety of XX. This is a complex torus, and an abelian variety when XX is.

  • The quotient Pic(X)/Jac(X)\mathrm{Pic}(X)/\mathrm{Jac}(X) is called the Néron–Severi group of XX and denoted NS(X)\mathrm{NS}(X). This is a finitely generated free abelian group.

So we have a short exact sequence

0Jac(X)Pic(X)NS(X)0 0 \to \mathrm{Jac}(X) \to \mathrm{Pic}(X) \to \mathrm{NS}(X) \to 0

You should think of it this way:

  • The Néron–Severi group NS(X)\mathrm{NS}(X) describes the ‘discrete degrees of freedom’ required to choose a holomorphic complex line bundle over XX. That’s because it’s the group of connected components of Pic(X)\mathrm{Pic}(X). It’s a finitely generated free abelian group.

  • The Jacobian Jac(X)\mathrm{Jac}(X) describes the ‘continuous degrees of freedom’ required to choose a holomorphic complex line bundle over XX. That’s because it’s the identity component of Pic(X)\mathrm{Pic}(X). It’s a complex torus.

But even better:

  • NS(X)\mathrm{NS}(X) is naturally isomorphic to the subgroup of H 2(X,)H^2(X,\mathbb{Z}) coming from holomorphic complex line bundles — which we described in our answer to Question 1.

  • Jac(X)\mathrm{Jac}(X) is naturally isomorphic to the dual torus X *X^*, which we described in our answer to Question 2.

Putting all these facts together, we get this:

Appell–Humbert Theorem. If XX is a complex torus, we have a short exact sequence

0Jac(X)Pic(X)NS(X)0 0 \to \mathrm{Jac}(X) \to \mathrm{Pic}(X) \to \mathrm{NS}(X) \to 0

where

  • NS(X)\mathrm{NS}(X) is isomorphic to the abelian group whose elements are alternating bilinear forms A:L×LA \colon L \times L \to \mathbb{Z} whose real-linear extension A˜\tilde{A} obeys A˜(iv,iw)=A˜(v,w)\tilde{A}(i v, i w) = \tilde{A}(v,w). (The set of these becomes an abelian group under addition.)

and

  • Jac(X)\mathrm{Jac}(X) is isomorphic to the dual torus X *X^\ast.

The dimension of the complex torus Jac(X)\mathrm{Jac}(X) depends only on the dimension of XX, since it’s just the dual torus. But the rank of the free abelian group NS(X)\mathrm{NS}(X) depends on more than just the dimension of XX. Indeed, a generic complex torus has no holomorphic complex line bundles on them except those that are topologically trival. For such complex, NS(X)\mathrm{NS}(X) has just one element! But in general NS(X)\mathrm{NS}(X) is a subgroup of H 2(X,)H^2(X,\mathbb{Z}) whose rank depends on the complex structure of XX, not just its dimension. It’s bigger for complex tori where the lattice LX˜L \subseteq \tilde{X} gets along better with the complex structure on X˜\tilde{X}, and these are the ones worth focusing on.

There’s a lot more to say, and I’m proud of myself for having not said it. Maybe I’ll say more later. I’m having tons of fun looking at examples of the theorems I just stated.

Posted at March 13, 2022 10:59 PM UTC

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10 Comments & 4 Trackbacks

Mozibur

There’s a lot more to say, but I’m proud for having not said it.

I saw a paper of Witten’s recently, String Theory & Noncommutative Geometry on the Arxiv. And in the comments under the abstract, he said:

100 pages, sorry!

I think long papers ought not to be called papers, like a long short story is not called a short story but a novella …

Posted by: Mozibur Ullah on March 14, 2022 2:31 PM | Permalink | Reply to this

Re: naming of long papers

We do have words like “monograph”.

And “book”.

Electronic distribution has somewhat eroded these distinctions, since a 500-page book can go on the arXiv just as easily as a 30-page paper (or a 5-page “note”), but maybe we should make more of an effort to use them.

Posted by: Mike Shulman on March 21, 2022 7:05 AM | Permalink | Reply to this

Re: Holomorphic Line Bundles on Complex Tori

That “sorry” must’ve been Seiberg’s. Witten writes papers that long all the time, unapologetically.

Posted by: Allen Knutson on March 15, 2022 3:34 PM | Permalink | Reply to this

Re: Holomorphic Line Bundles on Complex Tori

A complex abelian variety is a group in the category of smooth complex projective varieties. They’re called that because — wonderfully — they turn out to all be abelian!

Sorry to comment on something so trivial, but is that really why they’re called abelian?

I ask because once upon a time, a long time ago, I went to the first couple of lectures of a course on abelian varieties, and I thought the lecturer began by saying something like “They’re not called abelian varieties because the group structure is abelian. But fortunately, it is abelian, otherwise the terminology would be very confusing”.

This is the only thing I retain from those lectures. If someone stopped me in the street and made me tell them everything I know about abelian varieties, I’d say “they turn out to be abelian groups, but that’s not why they’re called abelian! (Now please let me go.)” If it turns out that I misremembered even that, then what I retain falls to zero. So, I cling on to the hope that I’m right.

Posted by: Tom Leinster on March 15, 2022 9:41 PM | Permalink | Reply to this

Re: Holomorphic Line Bundles on Complex Tori

Hmm, I think you’re right: maybe the fact that they’re abelian groups in the category of varieties is not why they were originally called abelian varieties!

Like many other great mathematicians of the 1800s, Abel did a lot of work on elliptic integrals and their generalizations, which are now called abelian integrals. This was later systematized using abelian varieties.

Here’s what Wikipedia says:

The theory of abelian integrals originated with a paper by Abel published in 1841. This paper was written during his stay in Paris in 1826 and presented to Augustin-Louis Cauchy in October of the same year. This theory, later fully developed by others, was one of the crowning achievements of nineteenth century mathematics and has had a major impact on the development of modern mathematics. In more abstract and geometric language, it is contained in the concept of abelian variety, or more precisely in the way an algebraic curve can be mapped into an abelian varieties. Abelian integrals were later connected to the prominent mathematician David Hilbert’s 16th Problem, and they continue to be considered one of the foremost challenges to contemporary mathematical analysis.

A slick modern way of summarizing some of this work is described here:

Namely, at least if we work over \mathbb{C}, the category of abelian varieties is monadic over the category of pointed projective varieties.

This may at first seem similar to how the category of abelian groups is monadic over the category of pointed sets. But there’s a big difference: for pointed varieties, we get an idempotent monad. That is, being an abelian group is really just a property of a pointed projective variety, not an extra structure!!!

Posted by: John Baez on March 15, 2022 10:42 PM | Permalink | Reply to this

Re: Holomorphic Line Bundles on Complex Tori

I’m not a historian, but I think there might be a connection between abelian varieties being abelian, and the naming of abelian groups, which is better than coincidence.

Abelian functions (in the modern terminology) are meromorphic functions g\mathbb{C}^g \to \mathbb{C}, periodic with respect to a lattice L gL \subset \mathbb{C}^g of rank 2g2g. From a modern perspective, these are meromorphic functions on the quotient A:= g/LA:=\mathbb{C}^g/L. The quotient AA is an abelian variety.

For any positive integer nn, multiplication by nn is a degree n 2gn^{2g} map μ:AA\mu: A \to A. This map can be written as some very complicated algebraic transformation from abelian functions on AA to other abelian functions on AA. If we want to invert this transformation, then we need to solve a polynomial of degree n 2gn^{2g}. However, this polynomial is unusually nice: It’s abelian, of the form (/n) 2g(\mathbb{Z}/n \mathbb{Z})^{2g}. (I’m glossing over some details here; we need to assume that the kernel μ 1(0)\mu^{-1}(0) is defined over our ground field.)

Now, what Cox says is that Abel was studying polynomials whose Galois group was what we now call abelian. Why was he studying such equations? Here I don’t know, but it sure seems reasonable to guess that he was trying to invert the transformations of abelian functions. (And Cox’s quote at the bottom of page 144 suggests I am right.) Why does inverting transformations of abelian functions involve solving equations with abelian Galois group? From a modern perspective, because these transformations encode multiplication by nn in the abelian group AA.

So I think that abelian groups were named because they are the sort of Galois groups that coming up when inverting transformations of abelian functions, and the reason those Galois groups are abelian is because abelian varieties are abelian.

Posted by: David Speyer on March 17, 2022 7:41 PM | Permalink | Reply to this

Re: Holomorphic Line Bundles on Complex Tori (Part 1)

Here is a link to an article by Steve Kleiman that reviews some of the history of Abelian integrals and Abelian varieties.

Posted by: Jason Starr on March 22, 2022 11:56 AM | Permalink | Reply to this

Re: Holomorphic Line Bundles on Complex Tori (Part 1)

That looks fun, especially the first part!

We develop in detail most of the theory of the Picard scheme that Grothendieck sketched in two Bourbaki talks and in commentaries on them. Also, we review in brief much of the rest of the theory developed by Grothendieck and by others. But we begin with a twelve-page historical introduction, which traces the development of the ideas from Bernoulli to Grothendieck, and which may appeal to a wider audience.

Posted by: John Baez on March 22, 2022 11:28 PM | Permalink | Reply to this
Read the post Line Bundles on Complex Tori (Part 2)
Weblog: The n-Category Café
Excerpt: Here are many descriptions of the Néron--Severi group of a complex torus!
Tracked: March 22, 2022 4:33 AM

Re: Holomorphic Line Bundles on Complex Tori (Part 1)

If you want to waste a little time, try to fill in the missing details in the proof of the Appell-Humbert Theorem presented in Griffiths-Harris. It is possible to do with some group cohomology, but I have no idea how a student with no such background is expected to finish that proof. I sometimes run a seminar on mathematical writing, and I present that proof next to the proof from Mumford’s “Abelian varieties” as one of the lectures.

Posted by: Jason Starr on March 22, 2022 11:25 AM | Permalink | Reply to this

Re: Holomorphic Line Bundles on Complex Tori (Part 1)

I may give it a try! Personally I’m having a lot of fun learning this material from Birkenhake and Lange’s Complex Abelian Varieties, and they give a fairly low-tech proof of the Appell–Humbert theorem early on. Since I’m taking a lowbrow computational approach to this material, what matters most to me is the explicit construction of a holomorphic line bundle from a pair (H,χ)(H, \chi) where HNS(X)H \in NS(X) and χ\chi is a ‘semicharacter’ for HH. This gives a map from NS(X)×X *NS(X) \times X^\ast to Pic(X)Pic(X), and so far I’m less concerned with the proof that this is one-to-one and onto.

I like ‘abstract nonsense’, including group cohomology, but since I’ll never be a full-fledged algebraic geometer, I’m focusing on playing around with some specific highly symmetrical examples of abelian varieties, and line bundles on those.

Posted by: John Baez on March 22, 2022 11:24 PM | Permalink | Reply to this
Read the post Holomorphic Gerbes (Part 1)
Weblog: The n-Category Café
Excerpt: Can we generalize a bunch of basic results about Picard groups and Néron--Severi groups from line bundles to gerbes?
Tracked: March 27, 2022 10:17 PM
Read the post Holomorphic Gerbes (Part 2)
Weblog: The n-Category Café
Excerpt: Two basic results on the classification of holomorphic $n$-gerbes.
Tracked: April 11, 2022 12:42 AM
Read the post Conversations on Mathematics
Weblog: The n-Category Café
Excerpt: Videos of math conversations between John Baez and James Dolan.
Tracked: July 17, 2022 12:24 AM

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