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 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 features, fairly seeing that Mumford's paintings, has been a big device within the research of abelian forms and invertible sheaves on them. additionally, abelian forms play an important function within the geometric method of glossy algebraic quantity idea. during this e-book, Kempf has thinking about the analytic points of the geometry of abelian forms, instead of taking the choice algebraic or mathematics issues of view. His function 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 team. this permits him to provide unique effects in regards to the projective excellent of an abelian kind. In its precise dialogue of the cohomology of invertible sheaves, the publication accommodates fabric formerly came upon simply in study articles. additionally, numerous examples the place abelian forms come up in quite a few branches of geometry are given as a end of the booklet.

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Let V(aj) = Lu~ak + Lw;Jak k for 15,j 5,g. k Then R(J + fV) = (rj) + f(O'j) where (rf) = R(J) and (0';) = T(w,J)(V) . Let (rf) = (xj) + iCy;) and (O'j) = (rj) + i(sj) be the real and imaginary parts. By definition bk = - L [ext + fr{) + (y{ + fs{)(J + fV)] aj j = - [ L(x{ + y{J) + f(r{ + s{ V + s{ V + s{J)] aj . 2 Siegel Upper-half Space 61 ~)r{ +s{J)aj = - Ly{Vaj = - LY{(Lu}ak + LW}Ja k) j j j I I In tenns of matrices this gives R = -y . W . Thus in the particular case (Y = l)T(w,J)(V) = -U - iW where V(aj) = L ujak k +L wjbk .

Therefore the ratio 'Y( (7) / (7 is a holomorphic function which always has absolute value 1. 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.

Then by definition R(J + fV) = R(J) + a(W,J) (V) . Let V(aj) = Lu~ak + Lw;Jak k for 15,j 5,g. k Then R(J + fV) = (rj) + f(O'j) where (rf) = R(J) and (0';) = T(w,J)(V) . Let (rf) = (xj) + iCy;) and (O'j) = (rj) + i(sj) be the real and imaginary parts. By definition bk = - L [ext + fr{) + (y{ + fs{)(J + fV)] aj j = - [ L(x{ + y{J) + f(r{ + s{ V + s{ V + s{J)] aj . 2 Siegel Upper-half Space 61 ~)r{ +s{J)aj = - Ly{Vaj = - LY{(Lu}ak + LW}Ja k) j j j I I In tenns of matrices this gives R = -y . W . Thus in the particular case (Y = l)T(w,J)(V) = -U - iW where V(aj) = L ujak k +L wjbk .

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