Download PDF by Alexander Polishchuk: Abelian Varieties, Theta Functions and the Fourier Transform

By Alexander Polishchuk

ISBN-10: 0511063911

ISBN-13: 9780511063916

ISBN-10: 051154653X

ISBN-13: 9780511546532

ISBN-10: 0521808049

ISBN-13: 9780521808040

This booklet is a contemporary therapy of the idea of theta features within the context of algebraic geometry. the newness of its process lies within the systematic use of the Fourier-Mukai rework. Alexander Polishchuk starts off by means of discussing the classical conception of theta features from the point of view of the illustration concept of the Heisenberg team (in which the standard Fourier rework performs the popular role). He then exhibits that during the algebraic method of this thought (originally as a result of Mumford) the Fourier-Mukai remodel can frequently be used to simplify the prevailing proofs or to supply thoroughly new proofs of many vital theorems. This incisive quantity is for graduate scholars and researchers with powerful curiosity in algebraic geometry.

Show description

Read Online or Download Abelian Varieties, Theta Functions and the Fourier Transform PDF

Best algebraic geometry books

Download PDF by G. Komatsu: Complex Geometry

Offers the court cases of a global convention on advanced geometry and comparable themes, held in commemoration of the fiftieth anniversary of Osaka collage, Osaka, Japan. The textual content makes a speciality of the CR invariants, hyperbolic geometry, Yamabe-type difficulties, and harmonic maps.

Read e-book online Explicit Birational Geometry of 3-folds PDF

One of many major achievements of algebraic geometry during the last 20 years is the paintings of Mori and others extending minimum types and the Enriques-Kodaira category to 3-folds. This built-in suite of papers facilities round purposes of Mori idea to birational geometry. 4 of the papers (those via Pukhlikov, Fletcher, Corti, and the lengthy joint paper by means of Corti, Pukhlikov and Reid) figure out intimately the speculation of birational tension of Fano 3-folds.

Download PDF by Jean-Paul Brasselet, Jose Luis Cisneros-molina, David: Singularities I: Algebraic and Analytic Aspects:

This is often the 1st a part of the court cases of the assembly 'School and Workshop at the Geometry and Topology of Singularities', held in Cuernavaca, Mexico, from January eighth to twenty sixth of 2007, in occasion of the sixtieth Birthday of Le Dung Trang. This quantity includes fourteen state-of-the-art learn articles on algebraic and analytic features of singularities of areas and maps.

Additional resources for Abelian Varieties, Theta Functions and the Fourier Transform

Example text

Hence, for a generic θ this situation does not occur. The proof of the fact that : T → P N induces an embedding of tangent spaces goes along similar lines. Assume that a holomorphic tangent vector Dv0 at the point v0 ∈ V / maps to zero under . Then we have Dv0 f = c · f for all f ∈ T (3H, , α 3 ), where c ∈ C is a constant. Let us extend Dv0 to a constant holomorphic vector field D on V . Considering the above condition for the subset of f of the form θ (v − a)θ(v − b)θ(v + a + b), we deduce that D(log θ) is a linear function (with constant term) for every non-zero θ ∈ T (H, , α).

E2n be the basis of the lattice H 1 (T, Z), ∗ e1∗ , . . , e2n be the dual basis of H 1 (T ∨ , Z), where T ∨ is the dual torus. Show that the first Chern class of the Poincar´e bundle on T × T ∨ is given by 2n c1 (P) = i=1 ei ∧ ei∗ . 2 Representations of Heisenberg Groups I This chapter in an introduction to the representation theory of Heisenberg groups. This theory will be our principal tool in the study of theta functions. Throughout this chapter V is a real vector space with a fixed symplectic form E.

For given data (H, , α), a Lagrangian subspace L compatible with ( , α) does not always exist. 2. We will also show there that for every given (H, ) as above, there exists α and a Lagrangian subspace L compatible with ( , α). 4. Lefschetz Theorem The standard application of the theory of theta functions is the following theorem of Lefschetz. 8. Let L be a holomorphic line bundle on the complex torus T = V / with c1 (L) = E; then for n ≥ 3 global holomorphic sections of L n define an embedding of T as a complex submanifold into P N .

Download PDF sample

Abelian Varieties, Theta Functions and the Fourier Transform by Alexander Polishchuk


by Paul
4.3

Rated 4.16 of 5 – based on 43 votes