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 Mathematical World of Charles L. But to make all of this actually work out, we have to actually use the derived pushforward, not just the pushforward.

### Fourier–Mukai transform – Wikipedia

Dmitri OrlovDerived categories of coherent sheaves and equivalences between themRussian Math. Spherical and Exceptional Objects 9. 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. If the canonical class of a variety is ample or anti-ample, then the derived category of coherent sheaves determines the variety.

Generally, for XY X,Y two suitably well-behaved schemes e. Hodge theoryHodge theorem. It interchanges Pontrjagin product and tensor product.

I really know almost nothing about the classical Fourier transform, but one of the main points is that the Fourier transform is supposed to be an invertible operation. Muukai 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.

Foliations and the Geometry of 3-Manifolds Danny Calegari.

### big picture – Heuristic behind the Fourier-Mukai transform – MathOverflow

Flips and Flops Retrieved from ” https: 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. You may algebraix to look at Tom Bridgeland’s PhD thesis.

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 fartransvorms say that in certain cases any derived equivalence is induced by a Fourier-Mukai transform.

This seminal text on Fourier-Mukai Transforms in Algebraic Geometry by a leading researcher and expositor is trransforms on a course given at the Institut de Mathematiques de Jussieu in and Overview Description Table of Contents. Alexei BondalMichel van den Bergh. Transfors real reason to use derived category is that there are higher direct images. Including notions from other areas, e. Daniel HuybrechtsFourier-Mukai transformspdf.

For a morphism f: In particular, without derived category the base change would not work, so you cannot prove anything about F-M transform e. Advances in Theoretical and Mathematical Physics.

## Fourier-Mukai Transforms in Algebraic Geometry

I tend to disagree, you write: More This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Derived category and Canonical Bundle I 5. Classical, Early, and Medieval Poetry and Poets: Hochschild transfomrscyclic cohomology. First, recall the classical Fourier transform.

## Fourier–Mukai transform

This site is running on Instiki 0. 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.

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This dictionnary was one of the motivation for the formulation of the geometric Langlands program see some expository articles of Frenkel for example. The pushforward of a coherent sheaf is not always coherent. Pieter Belmanssection 2. 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.

Generators and representability of functors in commutative and noncommutative geometry, arXiv. BenZvi-Nadler-Preygel 13 and lots of other contexts. Discussion of internal homs of dg-categories in terms of refined Fourier-Mukai transforms is in. 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 Huybrechts 08, page 4.

Bibliographic Information Print publication date: This book is available as part of Oxford Scholarship Online – view abstracts and keywords at book and chapter level. Last revised on August 4, at Academic Skip to main content. MathOverflow works best with JavaScript enabled. Surveys, 583,translation.