Flops and derived categories.
type: Item
instance of: scholarly article
title: Flops and derived categories.
zbMATH Open document ID: 1085.14017
author: Tom Bridgeland
published in: Inventiones Mathematicae
publication date: 2002-08-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0009053
review text: In the present paper, the author gives a new interpretation of smooth threefold flops in terms of Fourier-Mukai transforms, thereby relating birational geometry to the theory of perverse coherent sheaves and their moduli. More precisely, let \(f: Y\to X\) be a birational morphism of projective varieties with 1-dimensional fibres and the property \({\mathbf R}f_*{\mathcal O}_X = {\mathcal O}_X\). Using techniques from the theory of triangulated categories, the author constructs a \(t\)-structure (à la Beilinson-Bernstein-Deligne) on the derived category \(D(Y)\) of coherent sheaves on \(Y\), which then gives rise to the category \(\text{Per}(Y/X)\) of relative perverse sheaves on \(Y\). An object \(F\) of \(\text{Per}(Y/X)\) is called a perverse point sheaf, iff \(F\) is numerically equivalent to the structure sheaf of a point \(y\in Y\), and the author's first main result establishes the fact that there is a projective scheme \({\mathcal M}(Y/X)\) representing the functor of equivalence classes of families of perverse point sheaves in \(\text{Per}(Y/X)\). Next it is shown how these fine moduli spaces \({\mathcal M}(Y/X)\) can be used to derive Fourier-Mukai type equivalences of derived categories. This is then applied to complex projective threefolds with terminal singularities, using the observation that crepant resolutions of terminal threefolds are related by a finite chain of flops, on the one hand, and that flops correspond to Fourier-Mukai transforms at the level of the involved derived categories of coherent sheaves, on the other. In this context, the author's second main theorem concerns birational geometry and reads as follows: If \(X\) is a complex projective threefold with terminal singularities and \[ \begin{matrix} Y_1 && Y_2\\ f_1\searrow&&\swarrow f_2\\ &X\end{matrix} \] are crepant resolutions, then there is an equivalence of derived categories of coherent sheaves \(D(Y_1)@>\sim>> D(Y_2)\). In particular, this theorem implies that birationally equiva…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J32
zbMATH DE Number: 1777234
zbMATH Keywords: birational geometry
zbMATH Keywords: threefolds
zbMATH Keywords: Calabi-Yau threefolds
zbMATH Keywords: coherent sheaves
zbMATH Keywords: perverse sheaves
zbMATH Keywords: moduli spaces
zbMATH Keywords: derived categories
Wikidata QID: Q60961511
MaRDI profile type: Publication
OpenAlex ID: W2001377460
arXiv ID: math/0009053
DOI: 10.1007/S002220100185
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description: scientific article