By Alexander Polishchuk

ISBN-10: 0521808049

ISBN-13: 9780521808040

This ebook is a latest therapy of the idea of theta services 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 way of discussing the classical idea of theta features from the point of view of the illustration conception of the Heisenberg staff (in which the standard Fourier remodel performs the popular role). He then exhibits that during the algebraic method of this concept (originally because of Mumford) the Fourier-Mukai rework can frequently be used to simplify the prevailing proofs or to supply thoroughly new proofs of many very important theorems. This incisive quantity is for graduate scholars and researchers with powerful curiosity in algebraic geometry.

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The ﬁrst assertion is clear. To prove (ii) we observe that the pairing , induces the surjective homomorphism A/I → I . Since the orders of 1 both groups are equal to |A| 2 , this map is an isomorphism. Now for every a ∈ A we have q(a + i) = i∈I a, i q(a), i∈I which is zero for a ∈ I . Hence, γ (q) = |I |−1 · 1 = 1. i∈I To prove (iii) let us consider the form q ⊕ q −1 on A ⊕ A. Applying (ii) to the diagonal subgroup {(a, a), a ∈ A} in A ⊕ A and using (i) we derive that γ (q) · γ (q −1 ) = γ (q ⊕ q −1 ) = 1.

3. If the Hermitian form H is not positive then the space Fock(V, J ) deﬁned above, is zero (see Exercise 7). Yet another model of the representation of H(V ) associated with the complex structure J strictly compatible with E, is the space F + (J ) = φ : H(V ) → C | φ(λh) = λφ(h), λ ∈ U (1); dφ PJl = 0; |φ|2 dv < ∞ , V where H(V ) acts by (hφ)(h ) = φ(h −1 h ), PJl denotes the left-invariant distribution of subspaces on H(V ), which is equal to 0 ⊕ PJ at the point (1, 0). The isomorphism between F + (J ) and F − (J ) is given by the correspondence φ(h) → φ(h −1 ).

Ii) Let ⊂ be a sublattice. Then α θ H, ,L α α(γ )−1 U(1,γ ) θ H, = γ ∈ /( + ∩L) ,L α U(α(γ ),γ ) θ H, = γ ∈ /( + ∩L) ,L . 32 Theta Functions I The proof is left to the reader. ” Remarks. 1. 3 of [18]. 2. 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.

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

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