By Alexander Polishchuk
This booklet is a contemporary therapy of the idea of theta features 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 means of discussing the classical conception of theta features 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 exhibits that during the algebraic method of this thought (originally as a result 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 powerful curiosity in algebraic geometry.
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Hence, for a generic θ this situation does not occur. The proof of the fact that : T → P N induces an embedding of tangent spaces goes along similar lines. Assume that a holomorphic tangent vector Dv0 at the point v0 ∈ V / maps to zero under . Then we have Dv0 f = c · f for all f ∈ T (3H, , α 3 ), where c ∈ C is a constant. Let us extend Dv0 to a constant holomorphic vector ﬁeld D on V . Considering the above condition for the subset of f of the form θ (v − a)θ(v − b)θ(v + a + b), we deduce that D(log θ) is a linear function (with constant term) for every non-zero θ ∈ T (H, , α).
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.
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. 8. Let L be a holomorphic line bundle on the complex torus T = V / with c1 (L) = E; then for n ≥ 3 global holomorphic sections of L n deﬁne an embedding of T as a complex submanifold into P N .
Abelian Varieties, Theta Functions and the Fourier Transform by Alexander Polishchuk