nLab
Jacobian variety

Context

Complex geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

To every nonsingular algebraic curve CC (over the complex numbers) of genus gg one associates the Jacobian variety or simply Jacobian J(C)J(C) either via differential 1-forms or equivalently via line bundles:

the Jacobian is the the moduli space of degree-00 line bundles over CC, i.e. the connected component

Jac(X)=Pic 0(X) Jac(X) = Pic_0(X)

of the identity of the Picard scheme of CC. 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 CJ(C)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 nn-dimensional XX

nnCalabi-Cau 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=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate 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.

Revised on August 26, 2014 05:04:17 by Urs Schreiber (82.113.106.36)