#### Volume 20, issue 3 (2016)

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Variation of Gieseker moduli spaces via quiver GIT

### Daniel Greb, Julius Ross and Matei Toma

Geometry & Topology 20 (2016) 1539–1610
##### Abstract

We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker stability. Under a boundedness assumption which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, we prove that semistable sheaves have a projective coarse moduli space that depends on a natural stability parameter. We then give two applications of this machinery. First, we show that given a real ample class $\omega \in {N}^{1}{\left(X\right)}_{ℝ}$ on a smooth projective threefold $X$ there exists a projective moduli space of sheaves that are Gieseker semistable with respect to $\omega$. Second, we prove that given any two ample line bundles on $X$ the corresponding Gieseker moduli spaces are related by Thaddeus flips.

##### Keywords
Gieseker stability, variation of moduli spaces, chamber structures, boundedness, moduli of quiver representations, semistable sheaves on Kähler manifolds
##### Mathematical Subject Classification 2010
Primary: 14D20, 14J60, 32G13
Secondary: 14L24, 16G20
##### Publication
Received: 26 September 2014
Revised: 5 June 2015
Accepted: 3 July 2015
Published: 4 July 2016
Proposed: Richard Thomas
Seconded: Jim Bryan, Frances Kirwan
##### Authors
 Daniel Greb Essener Seminar für Algebraische Geometrie und Arithmetik Fakultät für Mathematik Universität Duisburg-Essen D-45117 Essen Germany Julius Ross Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB UK Matei Toma Institut de Mathématiques Élie Cartan Université de Lorraine BP 70239 54506 Vandoeuvre-lès-Nancy Cedex France