Download Abelian Varieties, Theta Functions and the Fourier Transform by Alexander Polishchuk PDF
By Alexander Polishchuk
This e-book is a latest remedy 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 rework. Alexander Polishchuk starts off by way of discussing the classical concept of theta capabilities from the perspective of the illustration idea of the Heisenberg team (in which the standard Fourier rework performs the widespread role). He then indicates that during the algebraic method of this conception (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 vital theorems. This incisive quantity is for graduate scholars and researchers with robust curiosity in algebraic geometry.
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Additional info for Abelian Varieties, Theta Functions and the Fourier Transform
Therefore, its image in H/U (1) = K is an isotropic subgroup L ⊂ K . Thus, one can specify an isotropic subgroup L ⊂ H by giving an isotropic subgroup L ⊂ K , equipped with a continuous lifting homomorphism σ : L → H (such that L = σ (L)). 1. For every isotropic subgroup L ⊂ K there exists a continuous lifting homomorphism σ : L → H . Proof. We have to prove that the central extension 1 → U (1) → p−1 (L) → L → 1 in the category of commutative locally compact groups splits. But this follows from the Pontryagin duality: the dual exact sequence gives an extension of U (1) = Z, hence it splits.
This implies that c(L 1 , L 2 , L 3 ) = γ (q L 1 ,L 2 ,L 3 ) in this case. Now let L 1 , L 2 , L 3 be an arbitrary admissible triple of Lagrangian subgroups. We construct another Lagrangian subgroup L = L(L 1 , L 2 , L 3 ) by setting L(L 1 , L 2 , L 3 ) = L 1 ∩ L 2 + (L 1 + L 2 ) ∩ L 3 (the lifting L(L 1 , L 2 , L 3 ) → H is given by the formula l12 + l3 → σ (l12 )σ (l3 ), where l12 ∈ L 1 ∩ L 2 , l3 ∈ L 3 ). It is straighforward to check that L is Lagrangian and is compatible with L i for i = 1, 2, 3.
If T = C/ is a 1-dimensional complex torus (complex elliptic curve) then for every nondegenerate Z-valued symplectic form E on the corresponding Hermitian form H on C is nondegenerate. Changing the sign of E if necessary we can achieve that H is positive. Hence, every complex elliptic curve is projective. 2. Let T = V / be a complex torus and T = V / → T be a ﬁnite unramiﬁed covering of T corresponding to a sublattice ⊂ of ﬁnite index. Then T is a complex abelian variety if and only if T is. Indeed, any polarization of T is also a polarization of T .