Complex Abelian Varieties and Theta Functions
Abelian varieties are a natural generalization of elliptic curves to higher dimensions, whose geometry and classification are as rich in elegant results as in the one-dimensional ease. The use of theta functions, particularly since Mumford's work, has been an important tool in the study of abelian varieties and invertible sheaves on them. Also, abelian varieties play a significant role in the geometric approach to modern algebraic number theory. In this book, Kempf has focused on the analytic aspects of the geometry of abelian varieties, rather than taking the alternative algebraic or arithmetic points of view. His purpose is to provide an introduction to complex analytic geometry. Thus, he uses Hermitian geometry as much as possible. One distinguishing feature of Kempf's presentation is the systematic use of Mumford's theta group. This allows him to give precise results about the projective ideal of an abelian variety. In its detailed discussion of the cohomology of invertible sheaves, the book incorporates material previously found only in research articles. Also, several examples where abelian varieties arise in various branches of geometry are given as a conclusion of the book.
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The Cohomology of Complex Tori
The Algebra of the Theta Functions
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abelian variety action acts algebraic ample analytic apply assume basis bundle called canonical character Clearly cohomology compact complex linear complex structure complex torus compute condition consider constant construction contained coordinates Corollary corresponding decomposition define determined differential dimension divisor element equation equivalent Exercise factor finite fixed follows formula function given gives Hence Hermitian form Hodge homomorphism imaginary induces injective integral invariant invertible sheaf irreducible isogeny isometry isomorphism Ker H lattice Lemma manifold mapping matrix metric multiplication natural non-zero operator positive definite projective Proof properties Proposition prove Reas(W Recall reduced representation respect result sheaves space subgroup subset subspace surjective symmetric symplectic SymR(W Symz(L tangent space Theorem theory theta functions tori transformation trivial unique unitary usual vector write zero