Download PDF by George R. Kempf: Complex Abelian Varieties and Theta Functions

By George R. Kempf

ISBN-10: 0387531688

ISBN-13: 9780387531687

Abelian forms are a traditional generalization of elliptic curves to raised dimensions, whose geometry and category are as wealthy in dependent effects as within the one-dimensional ease. using theta capabilities, fairly considering the fact that Mumford's paintings, has been a huge software within the examine of abelian forms and invertible sheaves on them. additionally, abelian types play an important function within the geometric method of smooth algebraic quantity concept. during this booklet, Kempf has fascinated with 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 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 staff. this permits him to provide specified effects in regards to the projective perfect of an abelian style. In its exact dialogue of the cohomology of invertible sheaves, the e-book comprises fabric formerly came upon simply in study articles. additionally, numerous examples the place abelian forms come up in numerous branches of geometry are given as a end of the publication.

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Extra resources for Complex Abelian Varieties and Theta Functions

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13. • ,a g and b1 , ••. ,bg in A and B. )bj = - El

4. We will next work out how SymIR(W) acts on the bundle (Jg x Sg (in particular how it acts on Sg). In terms of our basis al," . ,ag , bl , ... 5. The action of SymlR(W) on X c g X Sg 9 matrices such that is given by [~ ~] *(Z,T)=(ZI,T')=C(CT+d)-lz,(aT+b)(cT+d)-I). Proof. Let A = [~ :]. Let J be the complex structure R- 1( T) on W. x = (W, A * J) (I,-r') -------+ (;g where the complex linear isomorphism X is given a complex 9 the same name. Therefore X(l, -T) = (1, -T') [~ ~] X 9 matrix of or rather (X, -X .

13. o Chapter 4. l' be a very ample invertible sheaf on an abelian variety X. l'. 1. l'}. Proof. l') is finite. Assume that Tx extends to a projective transformation. Then for any E in IDI the divisor E - x is linearly equivalent to E. l'). l'is an isomorphism, then the global effect of a defines an isomorphism IP n ~ IP n which extends the action of Tx on X. l') on IPn. l') but the purpose of this section is to lay the groundwork for this presentation. Let p: G ~ PGL(n) be a projective representation of a group G.

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