By Alexander Polishchuk
This booklet is a latest remedy of the idea of theta features 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 services from the perspective of the illustration thought of the Heisenberg workforce (in which the standard Fourier remodel performs the well-liked role). He then exhibits that during the algebraic method of this concept (originally as a result 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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Additional resources for Abelian Varieties, Theta Functions and the Fourier Transform
Note that the 1-dimensional subspace T (H, L , α) ⊂ T (H, , α) coincides with the space of I -invariants in T (H, , α), where I = ⊥ ∩ L/ ∩ L is a maximal isotropic subgroup in ⊥ / , lifted to the ﬁnite Heisenberg group G(E, , α) trivially. Thus, we obtain the following corollary. 6. The theta series θ H, ,L is a generator of the 1-dimensional I subspace T (H, , α) ⊂ T (H, , α). 4) is described in the following proposition. 7. 2). (ii) Let ⊂ be a sublattice. Then α θ H, ,L α α(γ )−1 U(1,γ ) θ H, = γ ∈ /( + ∩L) ,L α U(α(γ ),γ ) θ H, = γ ∈ /( + ∩L) ,L .
Indeed, we have ⊥ and l(v) = π H (δ, γ ) = l(γ ) + 2πim(γ ) for some homomorphism m : m : V → R we get → Z. Extending m to an R-linear map π H (δ, v) − l(v) = 2πim(v) for every v ∈ V . It follows that π H (v, δ) − l(v) = 2πi E(v, δ) + 2πim(v). But the LHS is C-linear and the RHS takes values in i R. It follows that both sides are zero which implies our claim. 2) θ(v + δ) = A exp π H (v, δ) + H (δ, δ) θ(v). 2 Set = + Zδ. We have seen that ⊂ ⊥ and the assumption δ ∈ implies that is stricly bigger than .
Here PJr denotes the right-invariant distribution of subspaces on H(V ), which is equal to 0 ⊕ PJ at the point (1, 0). The unitary structure on F − (J ) is given by φ1 , φ2 = φ1 φ2 dv V (F − (J ) is complete with respect to this metric). The second model is a slight modiﬁcation of the ﬁrst one. Its importance is due to the fact that it uses holomorphic functions on V with respect to the complex structure J . Deﬁnition. 1) 2 where (λ, v) ∈ H(V ), v ∈ V . The Hermitian form on Fock(V, J ) is given by f (v)g(v) exp(−π H (v, v))dv f, g = V (the space Fock(V, J ) is complete with respect to the corresponding norm).
Abelian Varieties, Theta Functions and the Fourier Transform by Alexander Polishchuk