Duality, Polarization and Riemann Form

Preliminaries: (Real) Bilinear Forms on Complex Vector Spaces

A complex vector space \(V=\mathbb{C}^g\) can be viewed as a real vector space of dimension \(2g\). Theorietically, we may consider all linear forms/bilinear forms on \(V\), however, some certain restriants should be introduced to maintain the complex structure. A skew-symmetric real bilinear form \(E:V\times V\rightarrow \mathbb{R}\) is said to be rotationally symmetric if \[ E(u,v)=E(iu,iv) \] where \(i=\sqrt{-1}\) in the induced linear transformation on \(V\) by the complex structure. Firstly note that for such a bilnear form \(E(\bullet,\bullet)\), the form \[E(i\bullet,\bullet)\] is symmetric. Moreover, from which we see that given \(\alpha,\beta\in\mathbb{R}\) satisfying \(\alpha^2+\beta^2=1\) we have \[\begin{aligned} E((\alpha+i\beta)\cdot u, (\alpha&+i\beta)\cdot v)=(\alpha^2+\beta^2)E(u,v)\\ &+E(\alpha u,i\beta v)+E(i\beta u,\alpha v)\\ &=E(u,v)+\alpha\beta(-E(iv,u)+E(iu,v))\\ &=E(u,v) \end{aligned}\] which is indeed the meaning of being rotational skew-symmetric if \(\alpha\) and \(\beta\) are equal to \(\cos\theta\) and \(\sin\theta\) for some \(\theta\in\mathbb{R}\) respectively.

Additionally, note that such rotationally symmetric forms correspond to Hermitian forms on \(V\) in an one-to-one mannar. To see this, to a rotational skew-symmetric form \(E\) associate \[ H(u,v)=E(iu,v)+iE(u,v) \] which is obviously a Hermitian form, and to a Hermitian form we may associate \[ \mathrm{Im}H \] which is clearly rotationally symmetric.

A lattice \(\Lambda\in V=\mathbb{C}^g\) is said to be full if \(\Lambda\otimes_\mathbb{Z}\mathbb{R}\rightarrow V\) is a surjection, which implies that \(\Lambda\) is free of rank \(2g\) if viewed as an abelian subgroup of \(V\). A Riemann form on a complex vector space \(V\) is the following data

A full lattice \(\Lambda\subset V\),
and a skew-symmetric form \(e:\Lambda\times\Lambda\rightarrow\mathbb{Z}\) such that the base change \(e\otimes\mathbb{R}\) is a rotationally symmetric form on \(V\) with the corresponding Hermitian form being positive-definite.

Complex Torus, Albanese variety

Given \(g\in\mathbb{N}_+\) and a full lattice \(\Lambda\in\mathbb{C}^g\), the quotient space \(\mathbb{C}^g/\Lambda\) is called a complex torus.

Let’s build the (iso-)morphism from a complex torus \(A\) to it’s Albanese variety.

There is a isomorphism \(u:\Lambda\rightarrow\mathrm{H}_1(A)\) defined as follows: for every point \(p\in\Lambda\) choose a path from the origin \(0\in\mathbb{C}^g\) to \(p\), which then correspondes to a cycle in \(A\) since \(0\) and \(p\) send to the same point of the torus \(A\).
A path in \(\mathbb{C}\) correspondes immediately to an object in \((\mathrm{H}^0(A,\Omega_{A/\mathbb{C}}^1))^*\), hence \(v:\mathbb{C}^g\rightarrow(\mathrm{H}^0(A,\Omega_{A/\mathbb{C}}^1))^*\) sends a point \(p\) to the object corresponding to a path linking \(0\) and \(p\).
It is clear that \(v(p)\) does not depend ot the choice of the path linking \(0\) and \(p\) since \(\mathrm{H}^0(A,\Omega_{A/\mathbb{C}}^1)\) consists only globally defined holomorphic forms. Since \(\mathbb{C}^g\) is a covering space (todo)
\(u\) and \(v\) are isomorphisms (todo)

Theorem of the Cube, Translation Invariance of a Line Bundle

The theorem of the cube I

Given a variety \(X\) and an abelian variety \(A\). Let \(f,g,h:X\rightarrow A\) be three morphisms , then for any \(\mathcal{L}\in\mathrm{Pic}(A)\) we have \[\begin{aligned} (f+g+h)^*\mathcal{L}&=(f+g)^*\mathcal{L}\otimes (g+h)^*\mathcal{L}\otimes (h+f)^*\mathcal{L}\\ &\otimes f^*\mathcal{L}^{-1}\otimes g^*\mathcal{L}^{-1}\otimes h^*\mathcal{L}^{-1}. \end{aligned}\]

The theorem of the cube II

Specifically, if \(X=A\) and \(x,y\in A\) are two points, then for any \(\mathcal{L}\in\mathrm{Pic}(A)\) we have

\[ t_{x+y}^*\mathcal{L}\otimes\mathcal{L}=t_x^*\mathcal{L}\otimes t_y^*\mathcal{L} \]

where \(t_a\) is the translation map \((+a)\) on \(A\). This is proved by setting \(f\), \(g\) and \(h\) to \(\mathrm{Id}_A\) and constant maps sending \(A\) to \(x\) and \(y\) respectively.

(todo: Translation invariance <=> Algebraically trivial)

Isogeny and Duality

Isogeny defines an equivalance relation.

To a complex torus \(V/\Lambda\),

Theorem (Riemann–Poincaré)

Let \(C\) be a compact Riemann surface (smooth projective curve) of genus \(g\), and let \(J(C)\) denote its Jacobian. Consider the Abel–Jacobi map: \[ \kappa^{(k)} : C^{(k)} \to J(C) \] from the \(k\)th symmetric product \(C^{(k)} := C^k/S_k\) into the Jacobian.

Let \(\theta \subset J(C)\) denote the theta divisor, and also write \(\theta\) for its cohomology class.

Then:

(Poincaré’s Formula) For \(k \le g\), the image \(\kappa^{(k)}(C^{(k)})\) represents the homology class \[ [\kappa^{(k)}(C^{(k)})] = \frac{1}{(g-k)!} \theta^{g-k} \in H_{2k}(J(C), \mathbb{Z}). \]
(Riemann’s Theorem) For \(k = g-1\), the image \(\kappa^{(g-1)}(C^{(g-1)})\) is a translate of the theta divisor: \[ \kappa^{(g-1)}(C^{(g-1)}) = \theta + (\text{constant}). \] Hence, the divisor \(\theta\) determines the canonical class of the curve via the Abel–Jacobi map.