# Complex Abelian Varieties and Theta Functions by George R. Kempf

By George R. Kempf

Abelian kinds are a usual generalization of elliptic curves to raised dimensions, whose geometry and category are as wealthy in stylish effects as within the one-dimensional ease. using theta features, quite considering the fact that Mumford's paintings, has been an immense software within the learn of abelian forms and invertible sheaves on them. additionally, abelian forms play an important function within the geometric method of sleek algebraic quantity idea. during this publication, Kempf has excited about the analytic features of the geometry of abelian forms, instead of taking the choice algebraic or mathematics issues of view. His objective is to supply an advent to advanced analytic geometry. therefore, he makes use of Hermitian geometry up to attainable. One distinguishing function of Kempf's presentation is the systematic use of Mumford's theta staff. this permits him to offer special effects concerning the projective perfect of an abelian style. In its distinct dialogue of the cohomology of invertible sheaves, the publication contains fabric formerly stumbled on merely in examine articles. additionally, a number of examples the place abelian kinds come up in quite a few branches of geometry are given as a end of the publication.

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Hence it is a constant u (locally) where lui = 1. Thus u = fJ-1a or rather ufJ = a as Y is connected. l'). l'. l'). l') ---+ 0 . 6. l') , we can find (x, a) with a Hennitian. 2 as Hennitian by verifying of the definitions. 1. l'). l'). l'). l'). l') is unitary. l') is EbEB(~) f(b)g(a). Thus the basis of delta functions {Ob} is a unitary basis of T. l') invariant. l') of the theta functions. l')-invariant. 4 The Isogeny Theorem up to a Constant 35 as we are dealing with irreducible representation PIe multiplies the length of vectors by a constant.

G]. The set of all such subsets is Ind. For I in Ind let dz I = dZi1 1\ ... 1\ dZi". Then an element of An has the form W = ElClnd wIdz i where WI is in AO. #I=n As with (a~. ) ) the operator 8 on differential forms sends An to An+l and 2 8 =-- O. We want an expression for the adjoint (8)* of 8. Consider the linear operator defined by (8)*(hdz I ) = El:s;d:5:#I( _l)dH ( a:' )* hdzI-{id}. 7. a) (8*) takes An into An-I. b) (8)* is adjoint to (8). c) ((8)*8 + 8(8)* )(/IdZI) = (E1 9 :5:g( a~, )*( a~,) + 7r EjEI hi)hdzI .

Let V = (;g with unit basis al, ... ,ago Let bj = Cj('E19~g Tfak) where cil .. ·Ic g is a successively divisible sequence of positive integers. Then as 1m Tf is non-singular L = ffiZla; ffi ffiZlb i is a lattice. We define a skew-symmetric form E on the real space of V by setting E( aj, ak) = E(bj , bk ) = 0 = E( a"~ bk) if 1 =I k and E(a/, b/) = c/. We want to write E as a matrix for the basis all ... ,a g , iall ... ,ia g • By the linear algebra this is t [1 o Re-T]-l [0-1 1m - T 1] [1 0 0 Re_T]-l 1m - T .