Framed sheaves over treefolds and symmetric obstruction theories
type: Item
instance of: scholarly article
title: Framed sheaves over treefolds and symmetric obstruction theories
zbMATH Open document ID: 1267.14072
author: Dragos Oprea
published in: Documenta Mathematica
publication date: 2013-04-17T00:00:00Z
full work available at URL: 11.html
review text: In the paper under review, the author studies moduli spaces of framed sheaves over local Calabi-Yau \(3-\)folds, and computes their Donaldson-Thomas Euler characteristics and topological Euler characteristics of the compactified moduli space of framed modules. If \(S\) is a smooth complex projective surface, we let \(X^{\circ}\) be the total space of the canonical bundle \(K_{S}\) of \(S\), which is an open Calabi-Yau \(3-\)fold, and \(\pi:X\longrightarrow S\) the projectivization of \(K_{S}+\mathcal{O}_{S}\), which is a compact \(3-\)fold carrying two divisors \(S_{\infty}\) (corresponding to the \(0-\)section of \(\mathcal{O}_{S}\)) and \(S_{0}\) (corresponding to the \(0-\)section of \(K_{S}\)), and we have \(X^{\circ}=X\setminus S_{\infty}\). The moduli space \(\mathfrak{M}_{n}\) of semistable framed modules \((E,\phi)\) of rank \(r\), trivial first and second Chern classes, \(\chi(E)=r\chi(\mathcal{O}_{S})-n\) and non-zero framing \(\phi:E\longrightarrow\mathcal{O}_{S_{\infty}}\otimes\mathbb{C}^{r}\) was constructed as a projective variety by \textit{D. Huybrechts} and \textit{M. Lehn} in [Int. J. Math. 6, No. 2, 297--324 (1995; Zbl 0865.14004)]. Here semistability is with respect to the ample divisor \(H=\pi^{*}H_{0}+\epsilon c_{1}(\mathcal{O}_{X}(1))\) (for some ample line bundle \(H_{0}\) on \(S\) and for \(\epsilon\in\mathbb{Q}_{+}\) sufficiently small), and with respect to a polynomial \(\delta\) of degree at most 2. Inside \(\mathfrak{M}_{n}\) there is the open subset \(\mathfrak{M}_{n}^{\circ}\) of framed sheaves, i. e. stable framed modules \((E,\phi)\) such that \(E\) is torsion free, locally free near \(S_{\infty}\) and \(\phi\) is an isomorphism along \(S_{\infty}\). In section 3 the author shows that \(\mathfrak{M}_{n}^{\circ}\) admits a symmetric perfect obstruction theory: an obstruction theory for \(\mathfrak{M}_{n}^{\circ}\) is a complex \(\mathbb{F}\) which can be associated to any universal family \((\mathcal{E},\Phi)\). Grothendieck duality a…
reviewed by: Arvid Perego
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 6155135
zbMATH Keywords: moduli spaces of sheaves
zbMATH Keywords: Donaldson-Thomas invariants
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6155135