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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This quantity is the 3rd of three in a sequence surveying the speculation of theta features which play a important function within the fields of advanced research, algebraic geometry, quantity concept and so much lately particle physics. in keeping with lectures given by way of the writer on the Tata Institute of primary study in Bombay, those volumes represent a scientific exposition of theta services, starting with their old roots as analytic features in a single variable (Volume I), pertaining to a number of the attractive methods they are often used to explain moduli areas (Volume II), and culminating in a methodical comparability of theta features in research, algebraic geometry, and illustration concept (Volume III).
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Extra resources for Abelian Varieties, Theta Functions and the Fourier Transform
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.