geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
To every nonsingular algebraic curve $C$ (over the complex numbers) of genus $g$ one associates the Jacobian variety or simply Jacobian $J(C)$ either via differential 1-forms or equivalently via line bundles:
the Jacobian is the moduli space of degree-$0$ line bundles over $C$, i.e. the connected component
of the identity of the Picard scheme of $C$. See also at intermediate Jacobian – Examples – Jacobian.
Jacobian varieties are the most important class of abelian varieties.
The Abel-Jacobi map $C\to J(C)$ is defined with help of periods …)
over the complex numbers, line bundles on a Jacobian variety over a given Riemann surface are naturally encoded by Riemann theta functions.
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
wikipedia Jacobian variety, Abel-Jacobi map
P. Griffiths, J. Harris, Principles of algebraic geometry
A. Beauville, Jacobiennes des courbes spectrales et systèmes Hamiltoniens complètement intégrables, Acta Math. 164 (1990), 211-235.
A generalizatioin of Abel-Jacobi map to the setting of formal deformation theory is in
Review for Riemann surfaces includes