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

By George R. Kempf

ISBN-10: 0387531688

ISBN-13: 9780387531687

Abelian kinds are a typical generalization of elliptic curves to raised dimensions, whose geometry and category are as wealthy in based effects as within the one-dimensional ease. using theta services, fairly considering Mumford's paintings, has been a tremendous instrument within the learn of abelian kinds and invertible sheaves on them. additionally, abelian forms play an important position within the geometric method of glossy algebraic quantity thought. during this publication, Kempf has curious about the analytic points of the geometry of abelian types, instead of taking the choice algebraic or mathematics issues of view. His goal is to supply an advent to complicated 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 enables him to offer exact effects concerning the projective excellent of an abelian sort. In its unique dialogue of the cohomology of invertible sheaves, the publication contains fabric formerly chanced on in simple terms in learn articles. additionally, numerous examples the place abelian types come up in a variety of branches of geometry are given as a end of the booklet.

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Extra resources for Complex Abelian Varieties and Theta Functions

Example text

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. £Oo). £11 2 (f,g). 4 The Isogeny Theorem up to a Constant Let f : X ~ Y be an isogeny of abelian varieties. Let M be an ample invertible sheaf on Y. * Jt is ample on X. * : T(Y, Jt) ~ T( X, ft') using a theory of theta functions of Jt and ft'. If we have a compatible (to be defined) decompositions (A'(ft'), B'(ft')) of H(ft') and A'(Jt), B'(Jt)) of H(Jt).

Assume that f is in AO. Then g(z + 1) = exp ( - 7r L hizjzj -7r( L hj(z;lj + zili + liTi)) )al iEN jEN x exp(7rH(z, 1) 7r + 2H(l, l))f(z) = a, exp (7r H( Z, 1) + iH(l, 1) )g( z) . So 9 is in AO(a,H). 6 Examples Let fi' be an invertible sheaf on a complex torus X = V / L. 10 (Riemann-Roch). The Euler characteristic x(fi') of fi' E(-l)idimHi(X,fi') is the intersection number Cl(g~)9 where 9 = dimX. Proof. Let fi' = fi'(a,H) be some Appel-Humbert data. Then Cl(fi') is the invariant two-form on X corresponding to the skew-symmetric fonn 1m H = E.

L' on an abelian variety. l') be the projection. l') is abelian. l'( a, k). l'). l'), C*) is injective. Hence it is an isomorphism because the dual group has the same order. l'), C*) is surjective. l'). l') on which C* acts by multiplication. l') which is good in sense that it is one on C. l'). l'). So h . l'( a, h)h. l'(7ra, 7rh)h. vx . l'). As the 32 Chapter 4. 2') Ck . l')-invariant subspace of V. 2') Ck· v x ' Clearly the action V is simply determined by our implicit choice of coset representatives but nothing else because if I .

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