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March 19, 2022

Line Bundles on Complex Tori (Part 2)

Posted by John Baez

Last time I explained how the job of classifying holomorphic line bundles LL on a complex torus XX breaks into two parts:

  • the ‘discrete part’: the underlying topological line bundle of LL is classified by an element of a finitely generated free abelian group called the Néron–Severi group NS(X)\mathrm{NS}(X).

  • the ‘continuous part’: the holomorphic line bundles with a given underlying topological line bundle are classified by a complex torus called the Jacobian Jac(X)\mathrm{Jac}(X).

Today I want to talk more about the discrete part: the Néron–Severi group. Studying examples of this leads to beautiful pictures like this one by Roice Nelson:

Before I can talk about these examples I want to go through some prerequisites, since I’m still learning them myself. In math you sometimes have to eat your vegetables of theory before you can have your dessert of examples!

Complex tori

The term complex torus actually means a couple of different things, but here I mean a compact connected complex Lie group. All these are automatically abelian, and they’re all of the form

X=V/L X = V / L

where VV is a complex vector space and LL is a lattice in VV. (Today all vector spaces will be finite-dimensional.)

Note that we can recover VV and LL knowing just XX as a complex Lie group, because the vector space VV is the universal cover X˜\tilde{X}, and the lattice LVL \subseteq V is the kernel of the projection

p:X˜X p \colon \tilde{X} \to X

So, a complex torus is really ‘the same’ as a complex vector space with a lattice in it! In fact there’s an equivalence between the category where

  • an object is a compact connected complex Lie group XX and a morphism is a complex Lie group homomorphism f:XXf: X \to X'

and the category where

  • an object is a pair (V,L)(V,L) consisting of a complex vector space VV and a lattice LVL \subseteq V, and a morphism f:(V,L)(V,L)f \colon (V,L) \to (V',L') is a linear map f:VVf: V \to V' that maps LL into LL'.

This is one reason that complex tori are worth studying! We’re really studying linear algebra enhanced with lattices: a wonderful blend of the continuous and the discrete. The same philosophy works for real tori, and if I were writing a textbook I’d start by working out the theory of those. But bringing the complex numbers into the game adds an extra level of subtlety. For example, all real tori of a given dimension are isomorphic, but this is far from true for complex tori, because there are different ways the lattice LVL \subseteq V can get along with multiplication by ii.

The Néron–Severi group

Last time I talked about the set Pic(X)\mathrm{Pic}(X) of isomorphism classes of holomorphic line bundles on our complex torus XX. This set Pic(X)\mathrm{Pic}(X) is an abelian group, because we can tensor line bundles. It’s called the Picard group of XX. And it’s a topological group, because there’s a way to say what it means for two holomorphic line bundles to be close to each other.

The Picard group has a subgroup Jac(X)\mathrm{Jac}(X), namely the connected component of the identity. It’s called the Jacobian of XX.

The Picard group also has a quotient group

NS(X)=Pic(X)Jac(X) \mathrm{NS}(X) = \frac{\mathrm{Pic}(X)}{\mathrm{Jac}(X)}

Its elements are the connected components of Pic(X)\mathrm{Pic}(X). It’s called the Néron–Severi group. It’s actually a finitely generated free abelian group!

So, the Picard group fits into 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

together with the Jacobian and the Néron–Severi group. I will focus on the latter today!

Last time I gave one rather concrete description of the Néron–Severi group. Today I want to give two more. The first was in terms of alternating bilinear forms. The second will be in terms of hermitian forms. And the third will be in terms of maps from the complex torus XX to its dual.

The first two descriptions take advantage of the philosophy that a complex torus is secretly a complex vector space with a lattice in it.

Description 1: alternating bilinear forms

The set of isomorphism classes of topological line bundles over XX forms a group under tensor product of line bundles, and this group is isomorphic to H 2(X,)H^2(X,\mathbb{Z}): the second cohomology with integer coefficients of XX as a topological space. We can identify the Néron–Severi group NS(X)NS(X) with the subgroup of H 2(X,)H^2(X,\mathbb{Z}) coming from holomorphic line bundles.

Using the fact that XX is a complex torus, one can show that 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}

where LL is the lattice associated to XX. So, NS(X)\mathrm{NS}(X) is some subgroup of this: some bunch of alternating bilinear maps closed under addition and subtraction. It’s defined using the fact that XX is a complex torus. Because XX is a complex torus, its universal cover V=X˜V = \tilde{X} is a complex vector space, and we can extend AA to an alternating real-bilinear map of vector spaces, which I’ll call by the same name:

A:V×V A \colon V \times V \to \mathbb{R}

The Néron–Severi group NS(X)NS(X) then turns out to consisting of alternating bilinear maps

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

whose extension obeys

A(iv,iw)=A(v,w) A(i v,i w) = A(v,w)

for all v,wVv,w \in V. In summary:

Theorem 11. The Néron–Severi group NS(X)NS(X) consists of alternating bilinear maps A:L×LA \colon L \times L \to \mathbb{Z} whose extension to VV is preserved by multiplication by ii.

Or equivalently:

Theorem 11'. The Néron–Severi group NS(X)NS(X) consists of alternating bilinear maps of vector spaces A:V×VA \colon V \times V \to \mathbb{R} that are integer-valued on the lattice LVL \subseteq V and preserved by multiplication by ii.

Description 2: hermitian forms

The inner product on a Hilbert space is sesquilinear: it’s conjugate-linear in the first variable and complex-linear in the second. For any alternating bilinear form

A:V×V A \colon V \times V \to \mathbb{R}

such that

A(iv,iw)=A(v,w) A(i v, i w) = A(v, w)

there’s a unique sesquilinear map

H:V×V H \colon V \times V \to \mathbb{C}

whose imaginary part is AA:

A(v,w)=ImH(v,w) A(v,w) = \mathrm{Im} H(v,w)

To show this, we just dream up a formula for HH, namely

H(v,w)=A(v,iw)+iA(v,w) H(v,w) = A(v, i w) + i A(v,w)

and check that it really is sesquilinear and its imaginary part is AA. We can also show that

H(w,v)=H(v,w)¯ H(w,v) = \overline{H(v,w)}

In this game, a sesquilinear form with this extra property is called a hermitian form. For example, the inner product on a complex Hilbert space is a hermitian form, but there are also hermitian forms that aren’t positive definite.

Conversely, any hermitian form HH has as its imaginary part an antisymmetric form.

Thus, there’s a correspondence between

  • antisymmetric real-valued bilinear forms AA with A(iv,iw)=A(v,w)A(i v, i w) = A(v, w)

and

  • hermitian forms HH

given by the condition that AA is the imaginary part of HH. And under this correspondence AA is in the Néron–Severi group, meaning it takes integer values on LL, if and only if the imaginary part of HH takes integer values on LL.

In short, we’ve shown:

Theorem 2. The Néron–Severi group NS(X)NS(X) consists of hermitian forms H:V×VH \colon V \times V \to \mathbb{C} whose imaginary part takes integer values on the lattice LVL \subseteq V.

This puts a new spin on things. As I mentioned, a complex torus is secretly the same thing as a complex vector space with a lattice in it. Now we’re seeing that the Néron–Severi group of the complex torus consists of hermitian forms on the complex vector space that take integral values on the lattice!

Description 3: maps from a torus to its dual

We can also go in the other direction and try to describe the Néron–Severi group using complex tori as much as possible, eliminating all mention of complex vector spaces and lattices. This approach is also used a lot.

The trick is to start with Theorem 2, and turn the hermitian form

H:V×V H \colon V \times V \to \mathbb{C}

into a linear map

h:V V * v H(,v) \begin{array}{rcl} h \colon V &\to& V^\ast \\ v &\mapsto & H(-,v) \end{array}

where V *V^\ast is — beware! — the space of conjugate-linear maps from VV to \mathbb{C}. That’s because HH is conjugate-linear in the first slot.

Now, you may or may not recall that last time I used V *V^\ast to mean the space of real-linear maps from VV to \mathbb{R}. Luckily there’s no deadly inconsistency in my notation here, because that is naturally isomorphic to the space of conjugate-linear maps from VV to \mathbb{C}! Indeed, one thing I had to realize while learning this subject is this:

Lemma. For any complex vector space VV the following three real vector spaces are naturally isomorphic:

  1. the space of real-linear map from VV to \mathbb{R}

  2. the space of complex-linear maps from VV to \mathbb{C}

  3. the space of conjugate-linear maps from VV to \mathbb{C}.

Proof. We can get from 2) or 3) to 1) by taking the real part. We can get from 2) to 3) or vice versa by taking the complex conjugate. So it’s enough to check is that we can get from 1) to 2). Here we use the fact that any real-linear map f:Vf\colon V \to \mathbb{R} extends uniquely to a complex-linear map g:Vg \colon V \to \mathbb{C} given by

g(v)=f(v)if(iv) g(v) = f(v) - i f(i v)

One can check that the real part of gg is ff, and that gg is complex-linear since g(iv)=ig(v)g(i v) = i g(v) (this is a little calculation).    ⬛

Next, suppose a hermitian form H:V×VH \colon V \times V \to \mathbb{C} lies in the Néron–Severi group as described in Theorem 2.

This means that H:L×LH \colon L \times L \to \mathbb{Z}, which is equivalent to the requirement that the corresponding map

h:V V * v H(,v) \begin{array}{rcl} h \colon V &\to& V^\ast \\ v &\mapsto & H(-,v) \end{array}

maps LL into its dual lattice:

L *={f:V:fisreallinearandf(v)forallvL} L^\ast = \{ f \colon V \to \mathbb{R} : f \; is\; real–linear \; and\; f(v) \in \mathbb{Z} \; for \; all \; v \in L \}

Thanks to the lemma, we can think of L *L^\ast as a subset of V *V^\ast. L *L^\ast as defined above consists of real-linear functionals, and our ‘official’ definition of V *V^\ast is that it consists of conjugate-linear functionals — but the Lemma bridges the gap.

This gives an alternative statement of Theorem 2:

Theorem 22'. The Néron–Severi group NS(X)NS(X) consists of linear maps h:VV *h \colon V \to V^\ast that map LL into L *L^\ast and have h *=hh^\ast = h.

The last condition, h *=hh^\ast = h, is a translation of this condition that a hermitian form H:V×VH \colon V \times V \to \mathbb{C} must satisfy: H(w,v)=H(v,w)¯H(w,v) = \overline{H(v,w)}.

Next we can translate this result into the language of complex tori. To do this we define the dual torus X *X^\ast by

X *=V */L * X^\ast = V^\ast/L^\ast

Note that a linear map h:VV *h \colon V \to V^\ast that maps LL into L *L^\ast induces a map from V/L=XV/L = X to V */L *=X *V^\ast/L^\ast = X^\ast, which I’ll call

θ:XX * \theta \colon X \to X^\ast

This is a map of complex tori. Conversely any map of complex tori θ:XX *\theta \colon X \to X^\ast gives rise to a map between their universal covers, which is a complex linear map

h:VV * h \colon V \to V^\ast

and this maps LL to L *L^\ast.

Using this correspondence, everything about Theorem 22' can be translated into the language of complex tori, and we get this:

Theorem 33. The Néron–Severi group NS(X)NS(X) consists of maps of complex tori θ:XX *\theta \colon X \to X^\ast such that θ *=θ\theta^\ast = \theta.

Here we use the fact that every map of complex tori f:XYf \colon X \to Y has an adjoint f *:Y *X *f^\ast \colon Y^\ast \to X^\ast. The condition θ *=θ\theta^\ast = \theta is the translation into the language of complex tori of the condition h *=hh^\ast = h in Theorem 22'.

That’s it, our third and final description of the Néron–Severi group!

The Rosati involution

I’m basically done, but I can’t resist pointing out that a trick we used in Theorem 3 has a name. Given a map of complex tori

θ:XX *\theta \colon X \to X^\ast

its adjoint is a map

θ *:XX *\theta^\ast \colon X \to X^\ast

In fact we have

(θ *) *=θ (\theta^\ast)^\ast = \theta

So taking the adjoint gives an involution

() *:hom(X,X *)hom(X,X *) (-)^\ast \colon hom(X,X^\ast) \to hom(X,X^\ast)

called the Rosati involution.

I’ve usually seen the Rosati involution described in a somewhat different way, which works only when XX is equipped with an extra structure called a ‘principal polarization’, which gives an isomorphism X *XX^\ast \cong X. That more common approach has a lot of advantages, and maybe I’ll talk about it someday! But we don’t really need it to state Theorem 3, unless I’ve made some sort of mistake here.

The Rosati involution is a wonderful thing. The Rosati Involution should be a title of a novel, like The Eiger Sanction or The Bourne Identity. But it’s also the key to understanding the Néron–Severi group more deeply, and getting into examples like this:

Posted at March 19, 2022 2:08 AM UTC

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Re: Line Bundles on Complex Tori (Part 2)

There are a number of calculations to check in my post. Just for completeness, and to while away the long and dreary evenings, I’d like to check them here.

Here’s one:

For any alternating bilinear form A:V×V A \colon V \times V \to \mathbb{R} such that A(iv,iw)=A(v,w) A(i v, i w) = A(v, w) there’s a unique sesquilinear map H:V×V H \colon V \times V \to \mathbb{C} whose imaginary part is AA: A(v,w)=ImH(v,w) A(v,w) = \mathrm{Im} H(v,w) To show this, we just dream up a formula for HH, namely H(v,w)=A(v,iw)+iA(v,w) H(v,w) = A(v, i w) + i A(v,w) and check that it really is sesquilinear and its imaginary part is AA. We can also show that H(w,v)=H(v,w)¯ H(w,v) = \overline{H(v,w)} In this game, a sesquilinear form with this extra property is called a hermitian form.

It’s obvious that ImH=AIm H = A, but let’s check that HH is a hermitian form. First let’s show it’s complex-linear in the second argument. Since it’s real-linear we just need to check

H(v,iw)=iH(v,w) H(v,i w) = i H(v, w)

So:

H(v,iw) = A(v,i 2w)+iA(v,iw) = A(v,w)+iA(v,iw) = i(A(v,iw)+iA(v,w)) = iH(v,w) \begin{array}{ccl} H(v,i w) &=& A(v, i^2 w) + i A(v, i w) \\ &=& - A(v,w) + i A(v, i w) \\ &=& i (A(v, i w) + i A(v,w)) \\ &=& i H(v,w) \end{array}

Given that it’s complex-linear in the second argument, HH will automatically be conjugate-linear in the first argument if we can show

H(w,v)=H(v,w)¯ H(w,v) = \overline{H(v,w)}

So let’s do that. Again we just compute:

H(w,v) = A(w,iv)+iA(w,v) = A(iv,w)iA(v,w) = A(v,iw)iA(v,w) = H(v,w)¯ \begin{array}{ccl} H(w,v) &=& A(w, i v) + i A(w,v) \\ &=& - A(i v, w) - i A(v,w) \\ &=& A(v, i w) - i A(v,w) \\ &=& \overline{H(v,w)} \end{array}

Here in the third step we finally used the fact that AA is preserved by multiplication by ii: this gives A(iv,w)=A(v,iw)A(i v, w) = -A(v, i w).

This was worth checking because it helped me catch a small flaw in my logic and fix my post.

Posted by: John Baez on March 21, 2022 8:08 AM | Permalink | Reply to this
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