BUNTES Abelian varieties over \(\CC\) (Alex)

Section 1.3 Abelian varieties over \(\CC\) (Alex)

The goal of this talk is to understand what abelian varieties look like over \(\CC\text{.}\) The goal for me is to understand what a (principal) polarisation is and why it is important.

First immediate question: why study complex theory at all? The most classical field, algebraically closed, archimidean, characteristic 0.

Recall/rapidly learn the picture for elliptic curves, given \(E\) an elliptic curve we have for some \(\Lambda\) a rank 2 lattice in \(\CC\)

\begin{align*} \CC/\Lambda \amp\xrightarrow{\sim} E(\CC) \subseteq \PP^2(\CC)\\ z\amp\mapsto (\wp(z) : \wp'(z) : 1)\\ 0\amp\mapsto (0 : 1 : 0) \end{align*}

where

\begin{equation*} \wp(z) = \frac{1}{z^2} + \sum_{\lambda \in \Lambda\smallsetminus\{0\}} \frac{1}{(z-\lambda)^2} - \frac{1}{\lambda^2}\text{.} \end{equation*}

This is a meromorphic function whose image lands in

\begin{equation*} y^2 = 4x^3 - g_2 x - g_3\text{.} \end{equation*}

So the \(\CC\) points of an elliptic curve are topologically a torus.

Subsection 1.3.1 Abelian varieties

Naturally one asks: does this generalise? Let \(A\) be an abelian variety over \(\CC\text{,}\) what does \(A(\CC)\) look like? Another torus?

Proposition 1.3.1.

\(A(\CC)\) is a compact, connected, complex lie group.

Proposition 1.3.2.

Let \(A\) be an abelian variety of dimension \(g\) over \(\CC\text{.}\) Then we have

\begin{equation*} A(\CC) \cong V/\Lambda \end{equation*}

where \(V\) is a \(g\) dimensional complex vector space and \(\Lambda\) is a full rank lattice of \(V\) (i.e \(\Lambda\) is a discrete subgroup of \(V\) s.t. \(\RR\otimes \Lambda = V\)).

Proof.

Differential geometry gives us a map of complex manifolds, the exponential map

\begin{equation*} \exp\colon\Tgt_0(A(\CC)) \to A(\CC) \end{equation*}

this is holomorphic. And since \(A(\CC)\) is abelian, this is a homomorphism also. In general this is locally an isomorphism around 0.

Claim: \(\exp\) is injective. There exists a neighborhood \(U\supseteq 0\) s.t. \(\exp(U) \cong U\text{.}\) Consider the image \(\exp(\Tgt_0 A(\CC))\text{.}\) For \(x\in \exp(\Tgt_0 A(\CC))\text{,}\) \(\{U+x\}\) are all open and give a cover. Thus \(\exp(\Tgt_0A(\CC))\) is open. Since \(A(\CC)\) is connected we are thus reduced to showing \(\exp(\Tgt_0 A(\CC))\) is closed also. Since \(\exp\) is a homomorphism, the image is a subgroup. So its complement is the union of its non-trivial cosets, which is open. Thus \(\exp(\Tgt_0A(\CC))\) is closed. Giving \(\exp(\Tgt_0A(\CC)) = A(\CC)\text{,}\) which proves the claim.

\(\exp\) is a local isomorphism, which gives that \(\ker(\exp)\) is discrete, i.e. a lattice. We now have

\begin{equation*} A(\CC) \cong \Tgt_0A(\CC)/ \ker(\exp) \end{equation*}

so as \(A(\CC)\) is compact we cannot have a kernel which is not full rank, as otherwise the quotient could not be compact.

Definition 1.3.3.

We call any such \(V/\Lambda\) a complex torus.

From the above isomorphism we can now read off properties of \(A(\CC)\) as a group.

Proposition 1.3.4.

\(A(\CC)\) is divisible, and \(A(\CC)\lb n\rb \cong (\ZZ/n\ZZ)^{2g}\text{.}\)

Proof.

\begin{equation*} A(\CC) \cong V/\Lambda \cong (\RR/\ZZ)^{2g} \end{equation*}

isomorphisms as groups, thus \(A(\CC)\) is divisible. Further, \((\RR/\ZZ)\lb n \rb = (\frac 1n \ZZ)/\ZZ\text{.}\)

Question: Given a complex torus \(V/\Lambda\text{,}\) does there exist an abelian variety \(A\) such that \(A(\CC) \cong V/\Lambda\text{?}\)

Example 1.3.5.

\begin{equation*} \CC/\Lambda \cong E(\CC) \text{ always in dim 1} \end{equation*}
\begin{equation*} \CC^2/\Lambda^2 \cong (E\times E)(\CC) \text{ sometimes yes in higher dimension} \end{equation*}
\begin{equation*} \CC^2/\langle (i, 0), (i\sqrt p, i), (1, 0), (0, 1)\rangle_\ZZ \end{equation*}
for \(p\) prime??? (I guess not, see Mumford)

Theorem 1.3.6. Chow.

If \(X\) is an analytic submanifold of \(\PP^n(\CC)\) then \(X\) is an algebraic subvariety.

By this theorem it is enough to analytically imbed \(V/\Lambda \hookrightarrow \PP^m\text{.}\) We can try and do this by mimicing the elliptic curve strategy, find enough functions \(\theta \colon V/\Lambda \to \CC\text{.}\)

Subsection 1.3.2 Cohomology

Proposition 1.3.7.

Let \(X = V/\Lambda\text{.}\) Then

\begin{equation*} H^r (X,\ZZ) \cong \{\text{alternating }r\text{-forms } \Lambda\times\cdots\times\Lambda\to \ZZ\}\text{.} \end{equation*}

Proof.

\(\pi\colon V\to V/\Lambda\) is a universal covering map, so

\begin{equation*} \Lambda = \pi^{-1} (0 ) \cong \pi_1(X,0)\text{.} \end{equation*}

Because all these spaces are nice

\begin{equation*} H^1 (X,\ZZ) \cong \Hom(\pi_1(X), \ZZ) \cong \Hom(\Lambda, \ZZ)\text{.} \end{equation*}

To extend to \(r \ne 1\) use the Künneth formula:

\begin{equation*} \xymatrix{ \bigwedge^r(H^1(X_1\times X_2, \ZZ)) \ar@{=}[r]\ar@{=}[d]^{\text{Künneth}} & H^r(X_1\times X_2 , \ZZ)\ar@{=}[dd]^{\text{Künneth}}\\ \bigwedge^r(H^1(X_1, \ZZ)\otimes H^1(X_2, \ZZ)) \ar@{=}[d]& \\ \bigoplus_{p+q=r}(\bigwedge^p(H^1(X_1, \ZZ))\otimes\bigwedge^q(H^1(X_2, \ZZ))) \ar@{=}[r] & \bigoplus_{p+q=r}(H^p(X_1, \ZZ)\otimes H^q(X_2, \ZZ)) } \end{equation*}

Since we know the proposition for \(S^1 = \RR/\ZZ\) by taking products and applying the above we get it for all complex tori \(V/\Lambda\text{.}\)

Proposition 1.3.8.

There is a correspondence

\begin{align*} \{\text{Hermitian forms }H \text{ on } V\} \amp\leftrightarrow\{\text{Alternating forms }E\colon V\times V\to\RR,\,E(iu,iv) = E(u,v)\}\\ H\amp\mapsto \im H\\ E(iu,v)+ i E(u,v) \amp\mapsfrom E\text{.} \end{align*}

Subsection 1.3.3 Line bundles

Now we will consider line bundles on \(X = V/\Lambda\text{,}\) that is

\begin{equation*} L\xrightarrow{\pi} X \end{equation*}

such that for any \(x\in X\) there exists \(U\ni x\) with \(\pi^{-1} (U) \cong \CC \times U\text{.}\) We can obtain these from hermitian forms and some auxiliary data as follows.

Definition 1.3.9.

If \(H\) is a hermitian form on \(V\) such that \(E(\Lambda\times\Lambda) \subseteq \ZZ\) there exists a map

\begin{equation*} \alpha \colon \Lambda \to \CC^*_1 = \{z\in \CC^* : |z| = 1\} \end{equation*}

such that

\begin{equation*} \alpha(u + v) = e^{i\pi E(u,v)} \alpha(u) \alpha(v)\text{.} \end{equation*}

Further, there is a line bundle \(L(H, \alpha)\) on \(X\) which is defined by quotienting \(\CC\times V\) by \(\Lambda\) which acts via

\begin{equation*} \phi_u(\lambda, v) = (\alpha(u)e^{\pi H(v,u) + \frac12 \pi H(u,u)}\lambda, v+u)\text{ for } u\in \Lambda\text{,} \end{equation*}

we'll denote by \(e_u\) the factor \(\alpha(u)e^{\pi H(v,u) + \frac12 \pi H(u,u)}\) for brevity.

Theorem 1.3.10. Appell-Humbert.

Any line bundle on \(X\) is of the form \(L(H,\alpha)\) for some \(H\text{,}\) \(\alpha\) as above. Further

\begin{equation*} L(H_1, \alpha_1) \otimes L(H_2, \alpha_2) = L(H_1+ H_2, \alpha_1\alpha_2)\text{.} \end{equation*}

In fact we have the following diagram

\begin{equation*} \xymatrix{ 0 \ar[r] & \Hom(\Lambda, \CC^*_1) \ar[r]\ar[d] & \{\text{data } (H,\alpha)\} \ar[r]\ar[d] & \{\text{gp. of Herm. } H \text{ w/ }E(\Lambda\times \Lambda) \subseteq \ZZ\} \ar[r]\ar[d] & 0 \\ 0 \ar[r] & \Pic^0(X) \ar[r] & \Pic(X) \ar[r]_c & \ker(H^2(X,\ZZ) \to H^2(X,\sheaf O_X)) \ar[r]& 0 } \end{equation*}

where \(\Pic(X)\) is the group of all line bundles on \(X\) and \(\Pic^0\) is the subgroup of those which are topologically trivial.

We wanted functions \(X\to \CC\text{.}\) Now we can instead consider sections \(s\) of \(L(H,\alpha) \xrightarrow{\pi} X\) i.e. maps \(s\colon X\to L(H,\alpha)\) with \(\pi\circ s = \id\text{.}\) Denote the space of such sections \(H^0(X,L(H,\alpha))\text{.}\)

Definition 1.3.11. Theta functions.

The sections of \(L(H, \alpha)\) correspond to holomorphic functions

\begin{equation*} \theta \colon V \to \CC \end{equation*}

such that \(\theta(z+ u) = e_u \theta(z)\text{,}\) we will call such a \(\theta\) a theta function for \((H,\alpha)\text{.}\)

If \(H\) is not positive definite the space of such functions is 0!

Proposition 1.3.12.

If \(H\) is positive definite, then the dimension of \(H^0(X, L(H,\alpha))\) is \(\sqrt{\det E}\) where we really mean the determinant of a matrix for \(E\) with respect to an integral basis.

Theorem 1.3.13. Lefschetz.

Given a positive definite \(H\text{,}\) there exists an imbedding \(X \hookrightarrow \PP^m\text{.}\)

Proof.

Sketch: Let \(L = L(H,\alpha)\text{,}\) consider \(L(H,\alpha)^{\otimes 3} = L(3H, \alpha^3)\text{,}\) take a basis of \(\theta_0,\ldots, \theta_d\) of \(H^0(X, L^{\otimes 3})\text{.}\)

Claim: \(\Theta\colon z\mapsto (\theta_0(z) :\cdots :\theta_d (z)) \subseteq \PP^d\) is an embedding.

To see that this is well defined, we must give a section of \(L^{\otimes 3}\) not vanishing at \(z\) for all \(z\in X\text{.}\) Let \(\theta \in H^0(X,L)\smallsetminus \{0\}\text{.}\) Then pick \(a,b\) such that the section of \(L^{\otimes 3}\) given by

\begin{equation*} \theta(z-a)\theta(z-b) \theta(z+ a+b) \end{equation*}

does not vanish. This is possible and thus we have a nonvanishing section of \(L^{\otimes 3}\text{.}\)

For injectivity, show that if the above section has the same values on \(z_1,z_2\) then it is a theta function for some sublattice. Almost all sections aren't theta functions for a sublattice (this uses Proposition 1.3.12).

Something similar must be done for tangent vectors.

Definition 1.3.14. Riemann forms.

A Riemann form is \(E\colon \Lambda \times \Lambda \to \ZZ\) alternating such that

\begin{equation*} E_\RR \colon V\times V\to \RR \end{equation*}

has the property that \(E(iu,iv) = E(u,v)\) and the corresponding Hermitian form is positive definite.

Definition 1.3.15. Polarizable tori.

A complex torus \(X = V/\Lambda\) is polarizable if there exists a Riemann form \(E\) on \(\Lambda\text{.}\)

Example 1.3.16. Proposition.

Every \(\CC/\Lambda\) where \(\Lambda = \langle 1,\tau\rangle_{\ZZ}\) is polarizable.

To see this take

\begin{equation*} E(u,v) = \frac{u\bar v}{\im \tau} \end{equation*}

as a Riemann form.

Putting everything together we have obtained an equivalence of categories

\begin{equation*} \{\text{abelian varieties over } \CC\} \leftrightarrow \{\text{polarizable complex tori}\}\text{.} \end{equation*}

Subsection 1.3.4 Isogenies

Definition 1.3.17. Isogenies of complex tori.

An isogeny of complex tori is a homomorphism \(V/\Lambda \to V'/\Lambda'\) with finite kernel.

Definition 1.3.18. Dual vector spaces.

Given \(V\) a complex vector space, let

\begin{equation*} V^* = \{f\colon V\to \CC : f(u+v) = f(u)+f(v),\,f(\alpha v) = \bar \alpha f(v) \} \end{equation*}

and given \(\Lambda \subset V\) a lattice, let

\begin{equation*} \Lambda^* = \{f\in V^* : f(\lambda) \in \ZZ\,\forall \lambda\in \Lambda\}\text{.} \end{equation*}

Definition 1.3.19. Dual tori.

If \(X = V/\Lambda\text{,}\) \(X^\vee = V^*/ \Lambda^*\) is the dual torus.

Proposition 1.3.20. Existence of Weil pairing.

\begin{equation*} X\times X^\vee \to \CC \end{equation*}

\begin{equation*} X[n] \times X^\vee[n] \to \left(\frac{1}{n^2} \ZZ/\frac{1}{n} \ZZ\right) \cong \ZZ/n\ZZ \end{equation*}

this is called the Weil pairing.

Can a complex torus be isogenous to its own dual? If \(X\) is polarizable then

\begin{align*} X \amp\to X^\vee\\ v\amp\mapsto H(v,-) \end{align*}

is an isogeny.

Definition 1.3.21.

A polarization is an isogeny \(X \to X^\vee\text{.}\)