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
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:
“There’s no art
To find the mind’s construction in the face:
He was a gentleman on whom I built
An absolute trust.”
—William Shakespeare (1564–1616)
“The money complex is the demonic, and the demonic is God’s ape; the money complex is therefore the heir to and substitute for the religious complex, an attempt to find God in things.”
—Norman O. Brown (b. 1913)
“At the end of every diet, the path curves back toward the trough.”
—Mason Cooley (b. 1927)