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

By Alexander Polishchuk

ISBN-10: 0521808049

ISBN-13: 9780521808040

This e-book is a contemporary remedy of the speculation of theta capabilities within the context of algebraic geometry. the newness of its procedure lies within the systematic use of the Fourier-Mukai rework. Alexander Polishchuk starts off through discussing the classical conception of theta capabilities from the point of view of the illustration idea of the Heisenberg team (in which the standard Fourier rework performs the famous role). He then indicates that during the algebraic method of this concept (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 robust curiosity in algebraic geometry.

Show description

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

Similar algebraic geometry books

Basic Algebraic Geometry 2 by Igor R. Shafarevich, Miles Reid PDF

Shafarevich's easy Algebraic Geometry has been a vintage and universally used creation to the topic seeing that its first visual appeal over forty years in the past. because the translator writes in a prefatory be aware, ``For all [advanced undergraduate and starting graduate] scholars, and for the numerous experts in different branches of math who desire a liberal schooling in algebraic geometry, Shafarevich’s e-book is a needs to.

Read e-book online Commutative Algebra And Algebraic Geometry: Joint PDF

The 1st Joint AMS-India arithmetic assembly used to be held in Bangalore (India). This booklet provides articles written through audio system from a different consultation on commutative algebra and algebraic geometry. incorporated are contributions from a few top researchers world wide during this topic quarter. the quantity includes new and unique learn papers and survey articles compatible for graduate scholars and researchers attracted to commutative algebra and algebraic geometry

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

Example text

4 Indeed, splitting the series for θ11 ( τ2 + 14 , τ ) into two series, the sum over Appendix A. Theta Series and Weierstrass Sigma Function 39 even n and the sum over odd n, we get 1 3πi 3 τ + ,τ = τ+ (−1)n exp πi 4n 2 + 4n + θ11 2 4 4 4 n + exp πi(2n + 1)2 τ + 2πi n + n 1 2 2τ + 1 2 3πi (τ + 1) . 4 It remains to note that the first sum is zero (as seen by substituting n → −n − 1). 3). 4) (1 − q n )2 n=1 where in the right-hand side we use multiplicative variables q = exp(2πiτ ), 1 u = exp(2πi z) (and where u 2 = exp(πi z)).

We will call a decomposition = 1 ⊕ 2 of the type described in the above proposition, an isotropic decomposition of . 2) for some c ∈ V which is uniquely determined modulo ⊥ . It is easy to see that the corresponding homomorphisms σα and σα are related as follows: σα (γ ) = (1, c)σα (γ )(1, c)−1 . 3) Therefore, we can define an isomorphism of the corresponding finite Heisenberg groups i c : G(E, , α) → G(E, , α ) : g → (1, c)g(1, c)−1 . 4) Now the operator U(1,c) (corresponding to the action of (1, c) on Fock representation) restricts to an isomorphism between T (H, , α) and T (H, , α ) compatible with the actions of G(E, , α) and G(E, , α ) via i c .

This is indeed true (see Exercise 4). Let us consider some examples of complex abelian varieties. Exercises 35 Examples. 1. If T = C/ is a 1-dimensional complex torus (complex elliptic curve) then for every nondegenerate Z-valued symplectic form E on the corresponding Hermitian form H on C is nondegenerate. Changing the sign of E if necessary we can achieve that H is positive. Hence, every complex elliptic curve is projective. 2. Let T = V / be a complex torus and T = V / → T be a finite unramified covering of T corresponding to a sublattice ⊂ of finite index.

Download PDF sample

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


by John
4.5

Rated 4.46 of 5 – based on 18 votes