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.

**Read Online or Download Complex Abelian Varieties and Theta Functions PDF**

**Similar algebraic geometry books**

**The Geometry of Moduli Spaces of Sheaves**

Now again in print, this very hot ebook has been up-to-date to mirror fresh advances within the thought of semistable coherent sheaves and their moduli areas, which come with moduli areas in optimistic attribute, moduli areas of crucial bundles and of complexes, Hilbert schemes of issues on surfaces, derived different types of coherent sheaves, and moduli areas of sheaves on Calabi-Yau threefolds.

**Spaces of Homotopy Self-Equivalences: A Survey**

This survey covers teams of homotopy self-equivalence sessions of topological areas, and the homotopy kind of areas of homotopy self-equivalences. For manifolds, the entire team of equivalences and the mapping type team are in comparison, as are the corresponding areas. incorporated are tools of calculation, a number of calculations, finite new release effects, Whitehead torsion and different components.

**Galois Theory of Difference Equations**

This e-book lays the algebraic foundations of a Galois conception of linear distinction equations and indicates its dating to the analytic challenge of discovering meromorphic features asymptotic to formal recommendations of distinction equations. Classically, this latter query was once attacked via Birkhoff and Tritzinsky and the current paintings corrects and tremendously generalizes their contributions.

- Positivity in algebraic geometry
- Tropical and Idempotent Mathematics: International Workshop TROPICAL-07 Tropical and Idempotent Mathematics August 25-30, 2007, Independent University ... J.-v. Ponnc
- Advances in Algebraic Geometry Motivated by Physics
- Complex Analytic Sets (Mathematics and its Applications)
- Topology, ergodic theory, real algebraic geometry: Rokhlin's memorial

**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 .

### Complex Abelian Varieties and Theta Functions by George R. Kempf

by James

4.0