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