By Alexander Polishchuk

ISBN-10: 0511063911

ISBN-13: 9780511063916

ISBN-10: 051154653X

ISBN-13: 9780511546532

ISBN-10: 0521808049

ISBN-13: 9780521808040

This e-book is a latest therapy of the idea of theta services within the context of algebraic geometry. the newness of its technique lies within the systematic use of the Fourier-Mukai remodel. Alexander Polishchuk begins by way of discussing the classical conception of theta services 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 indicates that during the algebraic method of this thought (originally because of Mumford) the Fourier-Mukai remodel can usually be used to simplify the present 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.

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

**Example text**

Namely, we set 1 F (L) = F(L) meas(K /L) 2 (morally, one should think of elements of F (L) as half-measures on K /L). Note that the space F (L) has a canonical Hermitian metric. Namely, for φ1 , φ2 ∈ F (L) the product φ1 φ2 descends to a measure on K /L, which then can be integrated. Deﬁnition. Lagrangian subgroups L 1 , L 2 ⊂ K equipped with liftings to H , are called compatible if L 1 + L 2 is a closed subgroup and their lifting homomorphisms to H agree on L 1 ∩ L 2 . Note that for compatible subgroups one has L 1 + L 2 = (L 1 ∩ L 2 )⊥ .

1. 1. Fourier Transform Let K be a locally compact abelian group, K be the Pointriagin dual group. As was already mentioned earlier, the reader will not lose much by assuming that K is either a real vector space or a ﬁnite abelian group. For every Haar measure µ on K the Fourier transform is the operator Sµ : L 2 (K ) → L 2 ( K ) given by ˆ = Sµ (φ)(k) ˆ k(k)φ(k)µ. k∈K In this situation there is a unique dual measure µ ˆ on K such that Sµ is unitary with respect to the Hermitian metrics deﬁned using µ and µ.

5. 1) where v ∈ V , γ ∈ . Deﬁnition. 1). Its elements are called canonical theta functions for (H, , α). 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.

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

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