Complex Abelian Varieties and Theta FunctionsAbelian 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. |
Contents
| 1 | |
The Existence of Sections of Sheaves | 9 |
Groups Acting on Complete Linear Systems | 29 |
Theta Functions | 37 |
4 | 39 |
3 | 49 |
Modular Forms | 69 |
4 | 76 |
The Linear System 2D | 86 |
Abelian Varieties Occurring in Nature | 94 |
Informal Discussions of Immediate Sources | 101 |
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action algebraic ample analytic Appel-Humbert assume b₁ b₂ C-analytic canonical basis cohomology compact complex linear complex structure complex torus complex torus V/L compute constant coordinates Corollary decomposition define differential divisor E(iv e₁ element endomorphism equation Exercise factors of automorphy given H is positive Hence Hermitian form Hermitian metric Hodge structure homomorphism Homz induces injective integral invariant invertible sheaf irreducible isogeny isometry isomorphism isotropic Kähler Kähler Manifold Ker H Ker(ƒ kernel L₁ lattice Lemma Let f manifold mapping matrix metric moduli morphism multiplication NA,J non-degenerate non-zero section polarized abelian varieties positive definite principal polarization projective proof of Theorem Proposition prove Reas(W sheaves skew-symmetric form subgroup subset subspace surjective symmetric symplectic Symz tangent space Theorem 2.1 theory of theta theta functions theta group tori transformation trivial unitary zero α β