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By Alexander Polishchuk

This publication is a latest remedy of the speculation 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 begins through discussing the classical conception of theta capabilities from the perspective of the illustration idea of the Heisenberg staff (in which the standard Fourier rework performs the sought after role). He then exhibits that during the algebraic method of this idea (originally as a result of Mumford) the Fourier-Mukai remodel can frequently be used to simplify the present 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

It is called the Fock representation (the corresponding representation space is the space of holomorphic functions on V ). For an isotropic lattice ⊂ V equipped with a lifting to a subgroup in H, the space of -invariants in Fock representation can be identified with the space of global sections of some holomorphic line bundle on the torus V / . 3 They will be studied in Chapters 3 and 5. For some parts of the theory of Heisenberg groups it is convenient to work with the category of locally compact abelian groups.

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 ), where m is the Maslov index vector space and c(L 1 , L 2 , L 3 ) = exp(− πim 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.

The complex structure on V can be recovered from PJ via the isomorphism V V ⊗R C/PJ . By the definition, a function f on V is J -holomorphic if and only if d f (PJ ) = 0, where PJ is extended to a translation-invariant complex distribution of subspaces on V . Now it is easy to check that E is compatible with a complex structure J if and only if the corresponding subspace PJ ⊂ V ⊗R C is isotropic with respect to E (extended to a C-bilinear form). Therefore, 0 ⊕ PJ is a Lie subalgebra in Lie(H(V )) ⊗R C = C ⊕ V ⊗R C.

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