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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Flips and Flops There is a case where the analogy between sheaves and functions is more than analogy: Most natural functors, including basic ones geomegry pushforwards and pullbacksare of this type. This site is running on Instiki 0. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. 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.

## Fourier–Mukai transform

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 gepmetry greatly explored recently. It interchanges Pontrjagin product and tensor product.

As it turns out — and this feature algebbraic pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety.

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. Ilya Nikokoshev 8, 9 60 Fourier-Mukai transforms always have left and right adjointsboth of which are also kernel transformations.

Hence this is a pull-tensor-push integral transform through the product correspondence. Tensor product of sheaves behave a lot like multiplication of functions Front Matter Title Pages Preface. This seminal text on Fourier-Mukai Transforms in Algebraic Geometry by a leading researcher and expositor is based on a geonetry given at the Institut de Mathematiques de Jussieu in and Sign up using Email and Password.

Alexei BondalMichel van den Bergh. A Quick Tour 3.

### Fourier-Mukai Transforms in Algebraic Geometry – Oxford Scholarship

Generally, for XY X,Y two suitably well-behaved schemes e. 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. Including notions from other areas, e. More This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view.

Ebook This title is available as an ebook. Introduction to Basic Homotopy Theory. 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. Huybrechts Abstract 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 II 7. Don’t have an account? Equivalence Criteria for Fourier-Mukai Transforms 8. This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. First, recall the classical Fourier transform.

### Fourier–Mukai transform – Wikipedia

Views Read Edit View history. Pullback of sheaves behave a lot like pullback of functions Aimed at postgraduate students with a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety.

Pieter Belmanssection 2. The real reason to use derived category is that there are higher direct images. Hodge theoryHodge theorem. I tend to disagree, you write: In particular, without derived category the base change would not work, so you cannot prove anything about F-M transform e.

The Mathematical World of Charles L. Overview Description Table of Contents. Classical, Early, and Medieval Prose and Writers: Spherical and Exceptional Objects 9.

Let g denote the dimension of X. Advances in Theoretical and Mathematical Physics. Note that the converse is not true: Introduction to Abstract Homotopy Theory.