# 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 .