Jacobi variety
The Jacobi variety is a complex -dimensional torus and is considered in function theory. The name goes back to the mathematician Carl Gustav Jacob Jacobi , who developed the theory of elliptical functions in which this variety plays an important role. This object finds particular application in Abel's theorem and in Jacob's inversion problem .
definition
Periodic grid
Let be a compact Riemann surface with gender and be the fundamental group of . Let it be a basis of the holomorphic differential forms . Then is called
the period lattice of .
Due to the linearity of the integral obtained immediately an additive group structure on . The period grid is a real grid .
Jacobi variety
As in the definition above, let it be a compact Riemann surface with gender and a base of . Then is called
Jacobi variety of .
properties
- Since both and have an additive group structure, one can understand the quotient of two groups. So it is algebraically a group of factors .
- But since there is also a lattice, one can understand it as a -dimensional torus, on which one can define a structure of a complex manifold .
- Taken together, the Jacobi variety is a Lie group .
literature
- Otto Forster : Riemann surfaces (= Heidelberg pocket books 184). Springer-Verlag, Berlin a. a. 1977, ISBN 0-387-08034-1 .