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.
Read or Download Abelian Varieties, Theta Functions and the Fourier Transform PDF
Similar algebraic geometry books
This variation has been referred to as ‘startlingly up-to-date’, and during this corrected moment printing you may be convinced that it’s much more contemporaneous. It surveys from a unified viewpoint either the fashionable kingdom and the tendencies of constant improvement in a variety of branches of quantity idea. Illuminated via easy difficulties, the primary principles of contemporary theories are laid naked.
From the studies of the 1st printing of this publication, released as quantity 6 of the Encyclopaedia of Mathematical Sciences: ". .. My normal effect is of a very great e-book, with a well-balanced bibliography, urged! "Medelingen van Het Wiskundig Genootschap, 1995". .. The authors supply the following an up-to-the-minute consultant to the subject and its major purposes, together with a few new effects.
This article presents an advent to ergodic conception appropriate for readers understanding uncomplicated degree conception. The mathematical necessities are summarized in bankruptcy zero. it's was hoping the reader could be able to take on learn papers after examining the booklet. the 1st a part of the textual content is worried with measure-preserving modifications of likelihood areas; recurrence homes, blending houses, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy concept are mentioned.
- New Trends in Algebraic Geometry (London Mathematical Society Lecture Note Series)
- Configuration Spaces over Hilbert Schemes and Applications (Lecture Notes in Mathematics)
- Chapters on algebraic surfaces
- Computational Algebraic Geometry (London Mathematical Society Student Texts)
- CRC Standard Curves and Surfaces with Mathematica, Third Edition (Advances in Applied Mathematics)
Extra info for Abelian Varieties, Theta Functions and the Fourier Transform
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 ) 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.