The Jacobian of a compact Riemann surface

Put $V=\mathrm{\Omega }(X{)}^{\ast }$ with its additive structure and let $\pi :V\to V\mathrm{/}\mathrm{\Lambda }$ be the coset projection. By [F1], $V$ is an abelian group; by [F4], $\mathrm{\Lambda }$ is a subgroup. Then [F10] gives the well-defined abelian quotient law $[\xi ]+[\eta ]=[\xi +\eta ]$, identity $[0]$, and inverse $[-\xi ]$. Equip this coset set with the quotient topology from [F11].

By [F5], for each homology class the functional $e(\gamma )$ is independent of the chosen symplectic basis and cycle representative, so its image $\mathrm{\Lambda }$ and the quotient equivalence relation are intrinsic to $X$. A change of complex basis changes the evaluation coordinates by an invertible complex-linear map; [F3] and [F12] make it a homeomorphism, so the quotient set, group, and topology are unchanged.

If $g=0$, [F2] and [F4] give $V=\mathrm{\Lambda }=0$ and the quotient is a point. If $g\ge 1$, under the inherited AC of [F18] use the normalized basis ${\omega }_1,\dots ,{\omega }_g$ from [F6] to define $\mathrm{\Phi }:V\to {\mathbb{C}}^g$ by $\mathrm{\Phi }(\xi )=(\xi ({\omega }_1),\dots ,\xi ({\omega }_g))$. By [F2] and [F3] this is a complex-linear isomorphism; [F6] gives $\mathrm{\Phi }(\mathrm{\Lambda })={\mathbb{Z}}^g+\mathrm{\Pi }{\mathbb{Z}}^g$ and a real basis of $V$ consisting of its period vectors, and [F7] identifies that full lattice with their integer span.

If $g=0$, step 3.1 is a point, hence compact. Assume $g\ge 1$ and write $v_1,\dots ,v_{2g}$ for the real basis of $V$ from step 3.1. By [F8], the half-open parallelotope $P=\{{\sum }_i{t}_i{v}_i:0<t_i\le 1\}$ has translates by $\mathrm{\Lambda }$ covering $V$, so $\pi (P)=V\mathrm{/}\mathrm{\Lambda }$. In basis coordinates its closure is the image of $[0,1{]}^{2g}$ under a linear isomorphism; that map is continuous by [F12], and [F13] makes the closure compact. Since $\pi$ is continuous, [F13] makes $\pi (\bar{P})=V\mathrm{/}\mathrm{\Lambda }$ compact.

If $g=0$, the quotient is a point and is Hausdorff, second-countable, and connected. Assume $g\ge 1$. By [F9], bounded subsets meet $\mathrm{\Lambda }$ finitely. For distinct classes $[\xi ]\mathrm{\ne }[\eta ]$, put $w=\xi -\eta \mathrm{\notin }\mathrm{\Lambda }$; the bounded set $\bar{B}(w,1)$ meets $\mathrm{\Lambda }$ in finitely many points, all at positive distance from $w$, while lattice points outside it are more than distance $1$ away. Hence some $r>0$ has $B(w,r)\cap \mathrm{\Lambda }=\mathrm{\varnothing }$. The projection is open by [F11], so $\pi (B(\xi ,r\mathrm{/}3))$ and $\pi (B(\eta ,r\mathrm{/}3))$ are disjoint open neighborhoods: an intersection would give a lattice difference in $B(w,2r\mathrm{/}3)$. Thus the quotient is Hausdorff. Since $\pi$ is open, these images of a countable rational-box basis form a countable basis: for any open quotient set and any point in it, choose a lift in its open preimage and a rational box around that lift contained in the preimage. The straight-line paths in [F14] also show $V$ and its quotient image are path-connected, hence the quotient is connected.

If $g=0$, $\pi :0\to 0$ is the identity covering, its deck group is trivial, and the single chart gives the complex atlas. Assume $g\ge 1$. The bounded-intersection property [F9] gives $ϵ>0$ with $B(0,ϵ)\cap \mathrm{\Lambda }=\{0\}$. For each $z\in V$, $U_z=B(z,ϵ\mathrm{/}3)$ has disjoint nonidentity translates, since an intersection with $U_z+\lambda$ would imply $\mathrm{\mid }\lambda \mathrm{\mid }<2ϵ\mathrm{/}3$. The translations are homeomorphisms, so [F15] makes the orbit projection a covering with deck group exactly those translations; the projection is open by [F11], and its restriction to $U_z$ is a homeomorphism onto $\pi (U_z)$, giving a chart. On overlaps the two local lifts differ by a continuous $\mathrm{\Lambda }$-valued map, locally constant because $\mathrm{\Lambda }$ is discrete; every chart transition is therefore locally a translation and holomorphic by [F16].

In the charts of step 5.1, addition is locally $(z,w)\mapsto z+w+c$ and inversion is locally $z\mapsto -z+c$ for a fixed lattice vector $c$, so both are holomorphic by [F16]. If $g=1$, [F4] and [F6] give $\mathrm{\Lambda }=\mathbb{Z}+\mathrm{\Pi }\mathbb{Z}$ with $\mathrm{Im}\mathrm{\Pi }>0$, and [F17] identifies this with the oriented complex-lattice quotient. For $g=0$, step 3.1 gives the one-point zero-dimensional torus. Any other complex basis changes coordinates by an invertible complex-linear map carrying $\mathrm{\Lambda }$ to its coordinate image; by [F3], [F5], and [F12], it induces the biholomorphic presentation isomorphism in the Definition.