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 . ${\displaystyle g}$

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 ${\displaystyle X}$ ${\displaystyle g\geq 1}$ ${\displaystyle \pi _{1}(X)}$ ${\displaystyle X}$ ${\displaystyle \omega _{1},\ldots \omega _{g}\in \Omega (X)}$

${\displaystyle {\text{Per}}(\omega _{1},\ldots ,\omega _{g}):=\left\{\left(\int _{\alpha }\omega _{1},\ldots ,\int _{\alpha }\omega _{g}\right)\in \mathbb {C} ^{g}\ |\ [\alpha ]\in \pi _{1}(X)\right\}}$

the period lattice of . ${\displaystyle X}$

Due to the linearity of the integral obtained immediately an additive group structure on . The period grid is a real grid . ${\displaystyle {\text{Per}}(X)}$

Jacobi variety

As in the definition above, let it be a compact Riemann surface with gender and a base of . Then is called ${\displaystyle X}$ ${\displaystyle g}$ ${\displaystyle \omega _{1},\ldots ,\omega _{g}}$ ${\displaystyle \Omega (X)}$

${\displaystyle {\text{Jac(X)}}:=\mathbb {C} ^{g}/{\text{Per}}(\omega _{1},\ldots ,\omega _{g})}$

Jacobi variety of . ${\displaystyle X}$

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 . ${\displaystyle {\text{Per}}(X)}$ ${\displaystyle \mathbb {C} ^{g}}$ ${\displaystyle {\text{Jac}}(X)}$
• 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 . ${\displaystyle {\text{Per}}(X)}$ ${\displaystyle {\text{Jac}}(X)}$ ${\displaystyle g}$
• 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 .