By Alexander Polishchuk

ISBN-10: 0511063911

ISBN-13: 9780511063916

ISBN-10: 051154653X

ISBN-13: 9780511546532

ISBN-10: 0521808049

ISBN-13: 9780521808040

This publication is a contemporary remedy of the speculation of theta services 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 by means of discussing the classical idea of theta capabilities from the point of view of the illustration conception of the Heisenberg team (in which the standard Fourier remodel performs the famous role). He then indicates that during the algebraic method of this concept (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 vital theorems. This incisive quantity is for graduate scholars and researchers with powerful curiosity in algebraic geometry.

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**Additional info for Abelian Varieties, Theta Functions and the Fourier Transform**

**Example text**

I) Since ( + ⊂ ⊥ L ⊂ ⊥ , ∩ L)⊥ = ,L L ⊥ α = θ H, L ,L . is a lattice. One has ∩ ( + L) = + ⊥ ∩ L. (ii) For an element γ + l ∈ L , where γ ∈ , l ∈ ⊥ ∩ L, set α(γ + l) = α(γ ). 2). (iii) This follows immediately from the deﬁnition, since L / L ∩ L = / ∩ L. 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.

This proposition implies the following result for which we will give an independent proof in Chapter 7. 6. If the lattice is self-dual (and H is positive-deﬁnite), then T (H, , α) is 1-dimensional. Remark. We obtained above a description of the (projectivization of the) spaces of global sections of line bundles L(H, α) in terms that are independent of complex structure. This description relied heavily on the fact that the Hermitian form H is positive. It is easy to see that if H is nondegenerate but not positive then L(H, α) has no holomorphic sections.

Let N ( ) be the normalizer of in H(V ). Then the group G(E, , α) := N ( )/ acts naturally on the space T (H, , α). 1, G(E, , α) is a Heisenberg group. More precisely, it is a central extension of the ﬁnite abelian group ⊥ / by U (1), where ⊥ = {v ∈ V : E(v, ) ⊂ Z}. Equivalent way to deﬁne an action of G(E, , α) on T (H, , α) is the following. 5). The restriction of this action to N ( ) descends to an action of G(E, , α) on the line bundle L(H, α −1 ) over V / , compatible with the action of ⊥ / on V / by translations.

### Abelian Varieties, Theta Functions and the Fourier Transform by Alexander Polishchuk

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