This seminal text on Fourier-Mukai Transforms in Algebraic Geometry by a leading researcher and expositor is based on a course given at the. Fourier-Mukai transforms in algebraic geometry. CHTS. Mathematisches Institut Universitat Bonn. CLARENDON PRESS • OXFORD. In algebraic geometry, a Fourier–Mukai transform ΦK is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which is.
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The fact that the function associated to the Fourier-Deligne transform of a sheaf is the usual Fourier transform of the function associated to the sheaf is a consequence of the Grothendieck trace formula. Derived Category and Canonical bundle II 7. Huybrechts 08, page 4. Oxford Scholarship Online This book is available as part of Oxford Scholarship Online – view abstracts teansforms keywords at book and chapter level.
Search my Subject Specializations: First, recall the classical Fourier transform.
Discussion in the context of geometric Langlands duality is in. Let me give a rough picture of the Fourier-Mukai transform and how it resembles the classical situation.
There is a case where the analogy between sheaves and functions is more than analogy: Generators and representability of functors in commutative and noncommutative geometry, arXiv.
Pushforward of sheaves behave a lot like transforks of functions Foliations and the Geometry of 3-Manifolds Danny Calegari. Pieter Belmanssection 2. For a morphism f: I hope this gives you a better geometrg of what is going on, though I have to admit that I don’t know of any good heuristic idea behind, e.
Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on fougier smooth projective variety. To purchase, visit your preferred ebook provider.
Fourier-Mukai transform in nLab
Note that the converse is not true: I think Huybrechts’ book “Fourier-Mukai transforms in algebraic geometry” is a good book to look flurier. Moreover, could someone recommend a concise introduction to the subject? Lin Dec 27 ’09 at Choose your country or region Close. Pi-algebraspherical object and Pi A -algebra.
GMRA 1, 2 23 As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. You may also be interested in reading about Pontryagin dualitywhich is a version of the Fourier transform for locally compact abelian topological groups this is obviously quite similar, at least superficially, to Mukai’s result about abelian varieties.
Fourier-Mukai Transforms in Algebraic Geometry
End Matter References Index. Oxford University Press is a department of the University of Oxford. I think this was proven by Mukai.
What is the heuristic idea behind the Fourier-Mukai transform? Tensor product of sheaves behave a fourker like multiplication of functions And so we have to work with the derived categories. I second Kevin’s suggestion of Huybrechts’ book, but if you want to to look at something shorter first I recommend the notes by Hille and van den Bergh.
This page was last edited on 20 Septemberat Where to Go from Here References Index. This defines a correspondence sheaf-function compatible with all the analogies cited by Kevin. I believe you do alyebraic Fourier transform 4 times to get your original function back.
Introduction to Abstract Homotopy Theory. Remark Though theorem is stated there for F F admitting a gepmetry adjointit follows from Bondal-van den Bergh that every triangulated fully faithful functor admits a right adjoint automatically see e. Post as a guest Name.
Lin Dec 30 ’09 at This dictionnary was one of the motivation for the formulation of the geometric Langlands program see some expository articles of Frenkel for example. You may want to look at Tom Bridgeland’s PhD thesis.
The Fourier-Mukai transform in algebraic geometry gets its name because it at least superficially resembles the classical Fourier transform. Just as CommRing behaves a lot like Set opI think there is probably some kind of general phenomenon that sheaves or vector bundles behave a lot like functions, which is what’s happening here. What is the connection to the algebrauc Fourier transform? Fourier-Mukai transforms always have left and right adjointsboth of which are also kernel transformations.
Ilya Nikokoshev 8, 9 60