By George R. Kempf

Abelian forms are a average generalization of elliptic curves to raised dimensions, whose geometry and class are as wealthy in dependent effects as within the one-dimensional ease. using theta capabilities, quite considering that Mumford's paintings, has been a big instrument within the research of abelian kinds and invertible sheaves on them. additionally, abelian forms play an important function within the geometric method of smooth algebraic quantity thought. during this e-book, Kempf has enthusiastic about the analytic features of the geometry of abelian forms, instead of taking the choice algebraic or mathematics issues of view. His goal is to supply an creation to advanced analytic geometry. therefore, he makes use of Hermitian geometry up to attainable. One distinguishing characteristic of Kempf's presentation is the systematic use of Mumford's theta staff. this enables him to offer targeted effects in regards to the projective excellent of an abelian sort. In its certain dialogue of the cohomology of invertible sheaves, the publication contains fabric formerly came across merely in study articles. additionally, a number of examples the place abelian forms come up in a variety of branches of geometry are given as a end of the e-book.

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L') n N) ~ A'(Jt) and similarly with B. l')) but Ker(f) = MIL = AlAn L ffi BIB n L. This proves the result. 0 A~AnLffiB~/BnL and Finally we have a definitive isogeny theorem. 7. 6(Ob) B( Jt). (b')=b Proof. 7 we know this formula upto constant. (b')=oObl). 5 in the Fourier series expansion of Ob' the term e 7r / 2B (fJ,fJ) (constant) occurs only 00 and the constant is 1. it(oo) but this is e 7r / 2B (fJ,fJ). So C = 1. 0 We will later see that this innocent looking theorem is responsible for a meriad of relations between theta functions.

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 . Proof. 6. Also b) follows from the lemma and the explicit formula for 8 given by 8(hdzI) = El

Thus our projective representation p is determined by the ordinary representation p of the central extension H of G by C*. 30 Chapter 4. Groups Acting on Complete Linear Systems Now let ft' be an invertible sheaf on X, we want to construct the natural central extension H(ft') of K(ft') by C* together with a representation of H(ft') on T(X, ft') which will give above projective representation of K(ft') on ]pn when ft' is very ample. This group H(ft') is called the theta (or Heisenberg) group of ft'.