Moduli spaces of stable bundles on Calabi-Yau varieties and Donaldson-Thomas invariants
type: Item
instance of: scholarly article
title: Moduli spaces of stable bundles on Calabi-Yau varieties and Donaldson-Thomas invariants
zbMATH Open document ID: 1223.14020
published in: Journal of Geometry and Physics
publication date: 2011-10-11T00:00:00Z
review text: Given a smooth projective variety \(X\) with an ample line bundle, moduli spaces of Gieseker semistable sheaves on \(X\) are very interesting objects of study. If \(X\) is a surface, then quite a lot of results are present in the literature, but the investigation of fairly basic questions concerning, for example, the smoothness or the topological invariants of these moduli spaces on higher-dimensional varieties are, in general, difficult. The first goal of the paper under review is to investigate moduli spaces of sheaves on certain Calabi--Yau varieties, namely hypersurfaces of \(\mathbb{P}^1\times \mathbb{P}\), where \(\mathbb{P}\) is a projective bundle over \(\mathbb{P}^1\). The author proves that for certain choices of the ample line bundle \(L\) and Chern character \(c\), the moduli space of sheaves which are semistable with respect to \(L\) and have Chern character \(c\) is smooth and irreducible, in fact, isomorphic to a projective space resp.\ to some Quot-scheme. In any case, strictly semistable sheaves do not exist. The second goal of the paper is to present some explicit computations of Donaldson--Thomas (DT) invariants. Recall that DT-invariants roughly count stable sheaves on Calabi--Yau threefolds. The author uses the explicit description given in the first part, applies it to the three-dimensional case and derives the generating series of DT-invariants for these Calabi--Yau threefolds.
reviewed by: Pawel Sosna
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 5956427
zbMATH Keywords: moduli spaces of sheaves
zbMATH Keywords: Gieseker stability
zbMATH Keywords: Calabi-Yau varieties
zbMATH Keywords: Donaldson-Thomas invariants
author: Laura Costa
MaRDI profile type: Publication
full work available at URL: j.geomphys.2011.06.008
OpenAlex ID: W2076818888
cites work: A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on \(K3\) fibrations.
cites work: Gromov–Witten theory and Donaldson–Thomas theory, I
cites work: On Donaldson-Thomas invariants of threefold stacks and gerbes
cites work: Generalized Donaldson-Thomas invariants
cites work: Zero dimensional Donaldson--Thomas invariants of threefolds
cites work: Donaldson-Thomas invariants of certain Calabi-Yau 3-folds
cites work: Rationality of moduli spaces of vector bundles on rational surfaces
cites work: Moduli spaces of vector bundles on higher-dimensional varieties.
cites work: MODULI SPACES OF STABLE BUNDLES ON K3 FIBERED CALABI–YAU THREEFOLDS
cites work: STABLE RANK-2 BUNDLES ON CALABI–YAU MANIFOLDS
cites work: Moduli of stable sheaves. I
cites work: Moduli of stable sheaves. II
cites work: Q3128810
cites work: Symmetric obstruction theories and Hilbert schemes of points on threefolds
cites work: Donaldson-Thomas type invariants via microlocal geometry
DOI: 10.1016/J.GEOMPHYS.2011.06.008
description: scientific article; zbMATH DE number 5956427