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

By Alexander Polishchuk

This publication is a latest remedy of the idea of theta capabilities within the context of algebraic geometry. the newness of its strategy lies within the systematic use of the Fourier-Mukai rework. Alexander Polishchuk starts off by way of discussing the classical concept of theta capabilities from the point of view of the illustration thought of the Heisenberg workforce (in which the standard Fourier remodel performs the favourite role). He then exhibits that during the algebraic method of this thought (originally as a result of Mumford) the Fourier-Mukai remodel can usually 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 or Download Abelian Varieties, Theta Functions and the Fourier Transform PDF

Best algebraic geometry books

Undergraduate Algebraic Geometry (London Mathematical Society Student Texts, Volume 12)

Algebraic geometry is, basically, the examine of the answer of equations and occupies a critical place in natural arithmetic. With the minimal of must haves, Dr. Reid introduces the reader to the fundamental strategies of algebraic geometry, together with: airplane conics, cubics and the gang legislation, affine and projective types, and nonsingularity and size.

Fractured fractals and broken dreams: self-similar geometry through metric and measure

Fractal styles have emerged in lots of contexts, yet what precisely is a trend? How can one make specified the constructions mendacity inside of gadgets and the relationships among them? This ebook proposes new notions of coherent geometric constitution to supply a clean method of this usual box. It develops a brand new notion of self-similarity referred to as "BPI" or "big items of itself," which makes the sphere a lot more uncomplicated for individuals to go into.

Ramanujan's Lost Notebook: Part IV

​​​​In the spring of 1976, George Andrews of Pennsylvania nation collage visited the library at Trinity collage, Cambridge, to envision the papers of the overdue G. N. Watson. between those papers, Andrews chanced on a sheaf of 138 pages within the handwriting of Srinivasa Ramanujan. This manuscript used to be quickly distinctive, "Ramanujan's misplaced computer.

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

Sample text

The corresponding space F( ) can be identified with the space of L 2 -sections of certain line bundle on V / (see Exercise 1). Henceforward, referring to the above situation, we will say that a lifting homomorphism σα is given by the quadratic map α and will freely use the correspondence α → σα when discussing liftings of a lattice to the Heisenberg group. , if and only if the skew-symmetric form E| × is unimodular. We will refer to such lattices as self-dual. 4. , as a Lie subalgebra in Lie(H(V )) ⊗R C.

It follows that exp(π H (δ, γ )) = exp(l(γ )) for all γ ∈ . We claim that this can happen only if δ ∈ π H (v, δ). Indeed, we have ⊥ and l(v) = π H (δ, γ ) = l(γ ) + 2πim(γ ) for some homomorphism m : m : V → R we get → Z. Extending m to an R-linear map π H (δ, v) − l(v) = 2πim(v) for every v ∈ V . It follows that π H (v, δ) − l(v) = 2πi E(v, δ) + 2πim(v). But the LHS is C-linear and the RHS takes values in i R. It follows that both sides are zero which implies our claim. 1) as follows: π θ(v + δ) = A exp π H (v, δ) + H (δ, δ) θ(v).

2) coming from the Hermitian metric on L(H, α −1 ). 1) with the definition of the Fock representation we would like interpret the condition f ∈ T (H, , α) for a holomorphic function f on V as the invariance of f under operators Uα(γ ),γ for all γ ∈ (where is lifted to H using α). Since nonzero elements of T (H, , α) are not square-integrable, to achieve such an interpretation we have to extend operators Uλ,v to a larger space of holomorphic functions. The correct way to enlarge Fock(V, J ) is to consider the space Fock−∞ (V, J ) consisting of holomorphic functions f on V such that f (x) = O n x · exp π H (x, x) 2 for some n.

Download PDF sample

Rated 4.40 of 5 – based on 26 votes