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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Search my Subject Specializations: This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Journal of High Energy Physics. GMRA 1, 2 23 The pushforward of a coherent sheaf is not always coherent.

## Fourier-Mukai Transforms in Algebraic Geometry

The Mathematical World of Charles L. I think this is supposed to be analogous to the statement I made about the classical Fourier transform being invertible.

I tend to disagree, you write: I hope this gives you a better idea of what is going on, though I have to admit that I don’t know of any good heuristic idea behind, e. Overview Description Table of Contents. If X X is a moduli space of line bundles over a suitable algebraic curvethen a slight variant of the Fourier-Mukai transform is the geometric Langlands correspondence in the abelian case Frenkel 05, section 4.

From Wikipedia, the free encyclopedia. In particular, without derived category the base change would not work, so you cannot prove anything about F-M transform e. Discussion in the context of geometric Langlands duality is in.

### Fourier-Mukai transform in nLab

The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

The Fourier-Mukai transform in algebraic geometry gets its name because it at least superficially resembles the classical Fourier transform. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. Last revised on August 4, at Generators and representability of functors in commutative and noncommutative geometry, arXiv.

Choose your country or region Close. And of course because it was studied by Mukai. Classical, Early, and Medieval Plays and Playwrights: 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 By using this site, you agree to the Terms of Use and Privacy Policy. Csar Lozano Huerta 1, 1 15 Including notions from other areas, e.

MathOverflow works best with JavaScript enabled. This defines tgansforms correspondence sheaf-function compatible with all the analogies cited by Kevin. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Classical, Early, and Medieval World History: Hochschild cohomologycyclic cohomology. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use for details see www.

Including notions from other areas, e. Fourier-Mukai transforms always have left and right adjointsboth of which are also kernel transformations. It’s something like this: Flips and Flops I think this was proven by Mukai.

Let gometry give a rough picture of the Fourier-Mukai transform and how it resembles the classical situation. I think Huybrechts’ book “Fourier-Mukai transforms in algebraic geometry” is a good book to look at. Huybrechts Abstract This book provides a systematic exposition of gekmetry theory of Fourier-Mukai transforms from an algebro-geometric point of view.

Sign up or log in Sign up using Google. Lin Dec 27 ’09 at Generally, for XY X,Y two suitably well-behaved schemes e. Thanks, that looks very interesting.

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. Sign up using Email and Password.

The following answers might be useful: But to make all of this actually work out, we have to actually use the derived pushforward, not just the pushforward. However I don’t geoketry enough to say anything more than that.

I second Kevin’s suggestion of Huybrechts’ book, but if you want to to look at something shorter first I recommend the notes mukal Hille and van den Bergh. Introduction to Basic Homotopy Theory. Surveys, 583,translation. It was believed that theorem should be true for all triangulated functors e.

In string theory, T-duality short for target space dualitywhich relates two quantum field theories or string theories with different spacetime geometries, is closely related with the Fourier-Mukai transformation, a fact that has transgorms greatly explored recently.

It interchanges Pontrjagin product and tensor product. What is the heuristic idea behind the Fourier-Mukai transform?