Titlebook: Complex Abelian Varieties and Theta Functions; George R. Kempf Textbook 1991 Springer-Verlag Berlin Heidelberg 1991 Abelian varieties.Abel

沟通

Complex Tori,The lattice . in a real or complex finite dimensional vector space . is a discrete subgroup such that the quotient group . is compact. The lattice . is a free Abelian group of rank equal to the real dimension of . and the induced mapping . ⊗. ℝ → . is an isomorphism and conversely.

吝啬性

The Existence of Sections of Sheaves,Let (.) be A.-H. data for a complex torus .. Our objective is

最小

The Cohomology of Complex Tori,Let . be a lattice in a real vector space .. Then the quotient . is a real torus. Using a basis .,..., . of . we have an isomorphism . which sends a real .-vector (λ.,..., λ.) to the .-coset ∑λ.. Thus a real torus is topologically isomorphic to a product of circles. Consequently the algebraic topology of . is obvious.

Eviction

ERUPT

Theta Functions,Let . be a free abelian group of finite rank with a non-degenerate integral-valued skew-symmetric form . → ℤ. In this section we will develop the structure of the symplectic lattice ..

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枫树

Moduli Spaces,Let . be a real vector space of dimension 2. with a non-degenerate skew-symmetric form .. Recall the symplectic group Sym.(.) consists of all ℝ-linear transformations . of . such that . for all . and . in ..

mucous-membrane

Mappings to Abelian Varieties,Let . be a complex torus and let . be a smooth connected variety. Let . be a fixed point of .. Let . be an analytic mapping. Then we have the pull-back mapping of differentials . : . = .) → .).

isotope

Isometric

Abelian Varieties Occurring in Nature,Let . be a free abelian group of rank 2.. An elementary structure on . is a complex subspace . of .⊗. ℂ such that .⊗. ℂ = . where . is the linear mapping induced by complex conjugation in ℂ. Thus dim. . = dim. .. Let π. and π. denote projective of . ⊗. ℂ onto . and ..