By Alexander Polishchuk

This publication is a latest remedy 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 rework. Alexander Polishchuk begins through discussing the classical idea of theta services from the point of view of the illustration conception of the Heisenberg workforce (in which the standard Fourier remodel performs the favorite role). He then indicates that during the algebraic method of this idea (originally as a result 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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2). Let T be a complex torus, e1 , . . , e2n be the basis of the lattice H 1 (T, Z), ∗ e1∗ , . . , e2n be the dual basis of H 1 (T ∨ , Z), where T ∨ is the dual torus. Show that the ﬁrst Chern class of the Poincar´e bundle on T × T ∨ is given by 2n c1 (P) = i=1 ei ∧ ei∗ . 2 Representations of Heisenberg Groups I This chapter in an introduction to the representation theory of Heisenberg groups. This theory will be our principal tool in the study of theta functions. Throughout this chapter V is a real vector space with a ﬁxed symplectic form E.

The ﬁrst step in this direction was recently done by I. Zharkov (see [138]) who constructed a canonical cohomology class in H i ( , H 0 (V, O)), where H 0 (V, O) is the space of holomorphic functions on V . There remains a question, whether there exists an H(V )-submodule F−∞ ⊂ H 0 (V, O) such that the above class lies in H i ( , F−∞ ) and such that the projectivization of F−∞ does not depend on a choice of complex structure. Exercises 1. Let (V, E) be a symplectic vector space and ⊂ V be an isotropic lattice equipped with a lifting to the Heisenberg group H(V ).

However, in Chapter 4 we will construct explicitly intertwining operators between representations F(L) for some pairs of Lagrangian subgroups in K . 3. Real Heisenberg Group We are mainly interested in the classical real Heisenberg group H(V ) of a symplectic vector space (V, E). By the deﬁnition, H(V ) = U (1) × V and the group law is given by (λ, v) · (λ , v ) = (exp(πi E(v, v ))λλ , v + v ). Note that for a real Lagrangian subspace L ⊂ V there is a canonical lifting homomorphism L → H(V ) : l → (1, l).