Non-commutative width and Gopakumar-Vafa invariants
type: Item
instance of: scholarly article
title: Non-commutative width and Gopakumar-Vafa invariants
zbMATH Open document ID: 1348.14040
author: Yukinobu Toda
published in: Manuscripta Mathematica
publication date: 2015-11-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1411.1505
review text: In the paper under review, the author describes a relation between two types of invariants involving birational flopping contractions on smooth complex three-folds: the non-commutative width introduced by \textit{W. Donovan} and \textit{M. Wemyss} [Duke Math. J. 165, No. 8, 1397--1474 (2016; Zbl 1346.14031)] and the genus zero Gopakumar-Vafa invariants defined by \textit{S. Katz} [J. Differ. Geom. 79, No. 2, 185--195 (2008; Zbl 1142.32011)]. Let \(f:X\to Y\) be a birational flopping contraction from a smooth quasi-projective complex three-fold \(X\), contracting a single rational curve \(\mathbb P^1\cong C\subset X\) to a single point \(p\in Y\). The contraction algebra associated to \(f\), denoted by \(A_{\mathrm{con}}\), was introduced by Donovan-Wemyss in the above mentioned paper and it is a finite-dimensional algebra given by the universal non-commutative deformation algebra of the curve \(C\) in \(X\). The non-commutative width and the commutative width are then defined, respectively, as \[ \mathrm{wid}(C):=\dim_{\mathbb C} A_{\mathrm{con}}, \; \mathrm{cwid}(C):=\dim_{\mathbb C} A_{\mathrm{con}}^{\mathrm{ab}}. \] The paper's main result is Theorem 1.1, where the author shows the following formulas \[ \mathrm{wid}(C)=\sum_{j=1}^l j^2 \cdot n_j, \; \mathrm{cwid}(C)=n_1, \] where the \(n_j\)'s are the genus zero Gopakumar-Vafa invariants defined by Katz as virtual numbers of one dimensional stable sheaves on \(X\) and \(l\) is the scheme theoretic length of \(f^{-1}(p)\) at \(C\). The proof of the above theorem combines results from Donovan-Wemyss work with deformation theoretic arguments.
reviewed by: Margarida Melo
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6505848
zbMATH Keywords: non-commutative width
zbMATH Keywords: genus zero Gopakumar-Vafa invariants
zbMATH Keywords: birational flopping contraction
zbMATH Keywords: three-folds
MaRDI profile type: Publication
OpenAlex ID: W1528894264
arXiv ID: 1411.1505
cites work: Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds
cites work: Flops and derived categories.
cites work: Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities.
cites work: Noncommutative deformations and flops
cites work: Q5470192
cites work: Genus zero Gopakumar-Vafa invariants of contractible curves
cites work: Q5287413
cites work: Equivalences of derived categories and \(K3\) surfaces
cites work: Braid group actions on derived categories of coherent sheaves.
cites work: Stability conditions and crepant small resolutions
cites work: Three-dimensional flops and noncommutative rings
DOI: 10.1007/S00229-015-0760-8
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description: scientific article