By Alexander Polishchuk

This ebook is a contemporary therapy of the idea of theta services 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 features from the point of view of the illustration conception of the Heisenberg crew (in which the standard Fourier rework performs the renowned role). He then indicates that during the algebraic method of this concept (originally because of Mumford) the Fourier-Mukai remodel can frequently 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 robust curiosity in algebraic geometry.

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**Example text**

For an isotropic lattice ⊂ V equipped with a lifting to a subgroup in H, the space of -invariants in Fock representation can be identiﬁed with the space of global sections of some holomorphic line bundle on the torus V / . 3 They will be studied in Chapters 3 and 5. For some parts of the theory of Heisenberg groups it is convenient to work with the category of locally compact abelian groups. However, we will mostly care about two classes of such groups: real vector spaces and ﬁnite abelian groups, so the reader may choose to think only about locally compact abelian groups out of these two classes.

In the case when is not necessarily maximal isotropic, we equip the space of canonical theta functions with the structure of a representation of a ﬁnite Heisenberg group associated with , and show that it is irreducible. As a geometric application of theta functions we prove the theorem of Lefschetz stating that a complex torus V / can be embedded holomorphically into a projective space if and only if there exists a positive-deﬁnite Hermitian form H on V such that Im H takes integer values on .

A lattice such that E| × is integervalued. 1) → U (1) should satisfy α(γ1 + γ2 ) = exp(πi E(γ1 , γ2 ))α(γ1 )α(γ2 ). 2) on the data used in the construction of a line bundle on V / . 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.