\(D\)-equivalence and \(K\)-equivalence
type: Item
instance of: scholarly article
title: \(D\)-equivalence and \(K\)-equivalence
zbMATH Open document ID: 1056.14021
author: Yujiro Kawamata
published in: Journal of Differential Geometry
publication date: 2004-03-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0205287
review text: For a smooth projective variety \(X\), \(D(X):=D^b( \text{Coh} (X))\) denotes the derived category of bounded complexes of coherent sheaves on \(X\). Smooth projective varieties \(X\) and \(Y\) are said to be \(D\)-equivalent if \(D(X)\) and \(D(Y)\) are equivalent as triangulated categories, i.e. if there exists an equivalence \(D(X) \to D(Y)\) of categories which commutes with translations and which maps any triangle in \(D(X)\) to a triangle in \(D(Y)\). The notion of \(K\)-equivalence is defined by the following conditions: \(X\) and \(Y\) are birationally equivalent and there exists a smooth projective variety \(Z\) together with birational morphisms \(Z\to X\) and \(Z \to Y\) such that \(f^* (K_X)\) and \(g^* (K_Y)\) are linearly equivalent. The author states the following conjecture: Both notions coincide for birationally equivalent smooth projective varieties, i.e. in this case \(X\) and \(Y\) are \(K\)-equivalent iff they are \(D\)-equivalent. The question to reconstruct a variety from its derived category was studied by \textit{A. Bondal} and \textit{D. Orlov} [Compos. Math. 125, 327--344 (2001; Zbl 0994.18007)]. The author proves the following generalization of their theorem: If \(X\) and \(Y\) are \(D\)-equivalent, then the following holds: 0. \(\dim (X) = \dim (Y) \;\;(=:n)\). 1. If \(K_X\) (resp. \(-K_X\)) is nef, then \(K_Y\) (resp. \(-K_Y\)) is also nef and \(\nu (X) = \nu (Y)\) (resp. \(\nu (X,-K_X) = \nu (Y,-K_Y) \)). 2. If \(\kappa (X)=n\) or \(\kappa (X,-K_X)=n\) then \(X\) and \(Y\) are \(K\)-equivalent. Conjecture 1.5 states that for a given smooth projective variety \(X\) there exist (up to isomorphism) only finitely many smooth projective varieties \(Y\) which are \(D\)-equivalent to \(X\). An affirmative answer (extending a result of \textit{T. Bridgeland} and \textit{A. Maciocia} [Math. Z. 236, 677--697 (2001; Zbl 1081.14023)]) is given for the case of surfaces. The last section of the paper is dealing with singular varieties \(X\), especia…
reviewed by: Marko Roczen
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 2052875
zbMATH Keywords: derived category of coherent sheaves
zbMATH Keywords: \(\mathbb{Q}\)-factorial terminal singularity
Wikidata QID: Q115201684
MaRDI profile type: Publication
OpenAlex ID: W1788750319
arXiv ID: math/0205287
DOI: 10.4310/JDG/1090351323
description: scientific article