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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Let me give a rough picture of the Fourier-Mukai transform and how it resembles the classical situation.

If the canonical class of a variety is ample or anti-ample, then the derived category of coherent sheaves determines the variety. Derived Categories of Surfaces Post as a guest Name. Fourier-Mukai transform – a first example Intuition for Integral Transforms Fourier transform for dummies The last one has my sketch of an answer which I’ll post here once it gets better.

More This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. You may want to look at Tom Transdorms PhD thesis.

Huybrechts, author Mathematisches Institut, Universitaet Bonn.

### Fourier-Mukai transforms in algebraic geometry / D. Huybrechts – Details – Trove

There are some cool theorems of Orlov, I forget the precise statements but you can probably easily find them in any of the books suggested so farwhich say that in certain cases any derived equivalence is induced by a Fourier-Mukai transform.

Hopefully somebody else can say something about that. The Mathematical World of Charles L. Dmitri GeomteryDerived categories of coherent sheaves and equivalences between themRussian Math. I believe you do the Fourier transform 4 times to get your original function back.

Spherical and Exceptional Objects 9. It interchanges Pontrjagin product ib tensor product. Let g denote the dimension of X. Classical, Early, and Medieval Plays and Playwrights: Advances in Theoretical and Mathematical Physics.

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## Fourier-Mukai Transforms in Algebraic Geometry

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. What is the heuristic idea behind the Fourier-Mukai transform? Bibliographic Information Print publication date: Views Read Edit View history.

This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. I think this was proven by Mukai. Classical, Early, and Medieval Poetry and Poets: 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. Pushforward of sheaves behave a lot like integration of functions Civil War American History: Huybrechts Abstract This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view.

Most natural functors, including basic ones like pushforwards and pullbacksare of this type. Thanks, that looks very interesting. Hodge theoryHodge theorem. Home Questions Tags Users Unanswered.

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 Academic Skip to main content. Print Save Cite Email Share. Including notions from other areas, e.

The real reason to use derived category is that there are higher direct images.

## Fourier–Mukai transform

Flips and Flops Including notions from other areas, e. But to make all of this actually work out, we have to actually use nukai derived pushforward, not just the pushforward. From Wikipedia, the free encyclopedia. University Press Scholarship Online. Generators and representability of functors in commutative and noncommutative geometry, arXiv. Derived Categories of Coherent Sheaves 4. Heuristic behind the Fourier-Mukai transform Ask Question.

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Post Your Answer Discard By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies. It was believed that theorem should be true for all triangulated functors e.

I posted a question related to this here: Such concept of integral transform is rather general and may be considered also in derived algebraic geometry e. That equivalence is analogous to the classical Fourier transform that gives an isomorphism between tempered distributions on a finite-dimensional real vector space and its dual.

Hence this is a pull-tensor-push integral transform through the product correspondence. Don’t have an account?