By George R. Kempf

Abelian forms are a typical generalization of elliptic curves to better dimensions, whose geometry and category are as wealthy in dependent effects as within the one-dimensional ease. using theta capabilities, rather on the grounds that Mumford's paintings, has been a tremendous software within the learn of abelian types and invertible sheaves on them. additionally, abelian kinds play an important position within the geometric method of sleek algebraic quantity conception. during this booklet, Kempf has fascinated 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. hence, he makes use of Hermitian geometry up to attainable. One distinguishing characteristic of Kempf's presentation is the systematic use of Mumford's theta crew. this permits him to provide specific effects concerning the projective excellent of an abelian kind. In its special dialogue of the cohomology of invertible sheaves, the ebook contains fabric formerly discovered in basic terms in examine articles. additionally, a number of examples the place abelian types come up in a number of branches of geometry are given as a end of the ebook.

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Example text

Let A EB B be a decomposition of the lattice L with respect to the skew-symmetric form E = ImH. -H. data. The sheaf f£ = f£( a, H) is called excellent with respect to the decomposition A EB B = L, or just excellent if there is no confusion. The whole theory of theta function revolves around the special properties of excellent sheaves. Consider L~ = {v E VIE(v,l) E 71 for all 1 in L}. We have L~ = A ~ EB B~ where A = L ~ n A 0 1R and B = L ~ n B 0 IR. Thus K( f£) = L ~ f L is the direct sum A ~ fA EB B~ f B.

In fact we have a Hermitian structure on this bundle given by H J on the fiber over J. 2. B is a homogeneous Hermitian bundle over Reas(W) with respect to the group SymIR(W), In turns out that Reas(W) has the structure of a Hermitian symmetric space. In other words Reas(W) has a complex structure, Hermitian metric (all invariant Wlder SymIR(W)) and an holomorphic isometry

Now A' . B'(J)(u) (terms symmetric in m and n). Therefore A = e- 21riE(m,n). 3. If ft' is non-degenerate then the center of H(ft') is C*. 2 Representations of the Theta Group 31 Proof. l') we need to find another element =1= 1. l'( in, n) integer. By assumption we know that the form E is non-degenerate. We need to see that if E(m, n) is an integer for all n in MLihen is in L. This is easy. , ... ,hg be a basis for L. Then the dual basis is a basis for M. Hence M is contained in the span of the original basis (double dual basis) which is L.

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