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The Picard Variety of an Arbitrary Variety

In: Abelian Varieties

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  • Serge Lang

    (Yale University, Department of Mathematics)

Abstract

We are going to see in this chapter how one can prove the existence of the Picard variety of a variety V, complete and non-singular in codimension 1 in a simple manner, once one knows the existence of the Picard variety of an abelian variety. The Picard variety of V is derived functiorially from that of its Albanese variety, and we shall use this fact to get the theory of divisiorial correspondences on a product U x V. As a special case, we obtain the theory of correspondences on a curves, which gives us the Lefschetz fixed point formula. The group of cor- respondence classes of the curve is isomorphic to the group of an endomorphism corresponds to the characteristic polynomial of the representation of the endomorphism on the first homology group (in the classical case), and its trace is the trace of this representation. Combining the Lefschetz fixed point formula with the results of Chapter V, we obtain in a natural way the Riemann hypothesis for curves.

Suggested Citation

  • Serge Lang, 1983. "The Picard Variety of an Arbitrary Variety," Springer Books, in: Abelian Varieties, chapter 0, pages 147-177, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-8534-7_6
    DOI: 10.1007/978-1-4419-8534-7_6
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