Contents

complex geometry

cohomology

# Contents

## Idea

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

$Jac(X) = Pic_0(X)$

of the neutral element of the Picard scheme of $C$. See also at intermediate Jacobian – Examples – Jacobian.

Jacobian varieties are the most important class of abelian varieties.

## Properties

### Abel-Jacobi map

The Abel-Jacobi map $C\to J(C)$ is defined with help of periods.

### Line bundles and theta functions

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 $X$

$n$Calabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian

## References

• 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

Last revised on November 19, 2020 at 10:06:18. See the history of this page for a list of all contributions to it.