Jacobian Variety - Construction Over For Complex Curves

Construction Over For Complex Curves

Over the complex numbers, the Jacobian variety can be realized as the quotient space V/L, where V is the dual of the vector space of all global holomorphic differentials on C and L is the lattice of all elements of V of the form


\omega \mapsto \int_{\gamma} \omega

where γ is a closed path in C.

The Jacobian of a curve over an arbitrary field was constructed by Weil (1948) as part of his proof of the Riemann hypothesis for curves over a finite field.

The Abel-Jacobi theorem states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be identified with its Picard variety of degree 0 divisors modulo linear equivalence.

Read more about this topic:  Jacobian Variety

Famous quotes containing the words construction, complex and/or curves:

    The construction of life is at present in the power of facts far more than convictions.
    Walter Benjamin (1892–1940)

    It would be naive to think that peace and justice can be achieved easily. No set of rules or study of history will automatically resolve the problems.... However, with faith and perseverance,... complex problems in the past have been resolved in our search for justice and peace. They can be resolved in the future, provided, of course, that we can think of five new ways to measure the height of a tall building by using a barometer.
    Jimmy Carter (James Earl Carter, Jr.)

    One way to do it might be by making the scenery penetrate the automobile. A polished black sedan was a good subject, especially if parked at the intersection of a tree-bordered street and one of those heavyish spring skies whose bloated gray clouds and amoeba-shaped blotches of blue seem more physical than the reticent elms and effusive pavement. Now break the body of the car into separate curves and panels; then put it together in terms of reflections.
    Vladimir Nabokov (1899–1977)