By Alexander Polishchuk

This publication is a contemporary remedy of the idea of theta features within the context of algebraic geometry. the newness of its strategy lies within the systematic use of the Fourier-Mukai remodel. Alexander Polishchuk starts off via discussing the classical conception of theta services from the perspective of the illustration idea of the Heisenberg crew (in which the standard Fourier rework performs the trendy role). He then exhibits that during the algebraic method of this idea (originally because of Mumford) the Fourier-Mukai rework can frequently 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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**Example text**

We will refer to such lattices as self-dual. 4. , as a Lie subalgebra in Lie(H(V )) ⊗R C. Indeed, a complex structure on V can be speciﬁed by a complex subspace P ⊂ V ⊗R C of dimension 12 dimR V such that P ∩ V = 0. The correspondence goes as follows: given a complex structure J , the subspace PJ ⊂ V ⊗R C consists of elements v ⊗ 1 + J v ⊗ i, v ∈ V (this is precisely the subspace V from the canonical decomposition V ⊗R C = V ⊕ V ). The complex structure on V can be recovered from PJ via the isomorphism V V ⊗R C/PJ .

It is easy to see that if H is nondegenerate but not positive then L(H, α) has no holomorphic sections. As we will show in Chapter 7, in this case one has H j (T, L(H, α)) = 0 for j = i, while H i (T, L(H, α)) = 0, where i is the number of negative eigenvalues of H . Moreover, in the case when is self-dual, the space H i (T, L(H, α)) Exercises 25 is 1-dimensional. 4 for H i (T, L(H, α)). 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 .

In the former case the group A(L 1 , L 2 , L 3 ) is ﬁnite and c(L 1 , L 2 , L 3 ) is equal to the Gauss sum associated with q. In the latter case A(L 1 , L 2 , L 3 ) is a vector space and c(L 1 , L 2 , L 3 ) = exp(− πim ), where m is the Maslov index 4 of the triple (L 1 , L 2 , L 3 ), which is equal to the signature of the quadratic form −q. Gauss sums associated with quadratic forms on ﬁnite abelian groups will appear in the functional equation for theta functions. In this chapter we show that they are always given by 8th roots of unity.