By Alexander Polishchuk

This booklet is a latest 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 remodel. Alexander Polishchuk starts off via discussing the classical conception of theta capabilities from the point of view of the illustration idea of the Heisenberg crew (in which the standard Fourier remodel performs the favourite role). He then exhibits that during the algebraic method of this idea (originally because of Mumford) the Fourier-Mukai rework can usually 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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**Extra resources for Abelian Varieties, Theta Functions and the Fourier Transform**

**Example text**

Exercises 7. 15 Let E : × → Z/2Z be a skew-symmetric bilinear form modulo 2 (skew-symmetry means that E(γ , γ ) = 0 for every γ ∈ ). Prove that there exists a map f : → Z/2Z, such that E(γ1 , γ2 ) = f (γ1 + γ2 ) + f (γ1 ) + f (γ2 ). 8. 2). Let T be a complex torus, e1 , . . , e2n be the basis of the lattice H 1 (T, Z), ∗ be the dual basis of H 1 (T ∨ , Z), where T ∨ is the dual torus. e1∗ , . . , e2n Show that the ﬁrst Chern class of the Poincar´e bundle on T × T ∨ is given by 2n c1 (P) = i=1 ei ∧ ei∗ .

Let H be a ﬁnite Heisenberg group, W be its Schr¨odinger representation. Show that W ∗ ⊗ W is isomorphic as H × H -representation to the space of functions φ on H such that φ(zh) = zφ(h) for z ∈ U (1) with the H × H -action given by (h 1 , h 2 )φ(h) = φ(h −1 1 hh 2 ). (a) Let us identify the Lie algebra of U (1) with R in such a way that the exponential map Lie(U (1)) → U (1) is given by x → exp(2πi x) and consider the induced identiﬁcation of Lie(H(V )) with R ⊕ V (as vector spaces). Show that the distribution PJr on H(V ) can be described explicitly as follows: PJr 7.

The latter condition is equivalent to the condition that the Hermitian form H on V , such that Im H = E, is positive deﬁnite. We are going to construct several equivalent models for the unitary representation of H(V ) associated with a complex structure J . The ﬁrst model is the space F − (J ) = φ : H(V ) → C | φ(λh) = λ−1 φ(h), λ ∈ U (1); dφ PJr = 0; |φ|2 dv < ∞ , V where the action of H(V ) is given by (hφ)(h ) = φ(h h). Here PJr denotes the right-invariant distribution of subspaces on H(V ), which is equal to 0 ⊕ PJ at the point (1, 0).