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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

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 ω N1(X) on a smooth projective threefold X there exists a projective moduli space of sheaves that are Gieseker semistable with respect to ω. Second, we prove that given any two ample line bundles on X the corresponding Gieseker moduli spaces are related by Thaddeus flips.

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
Received: 26 September 2014
Revised: 5 June 2015
Accepted: 3 July 2015
Published: 4 July 2016
Proposed: Richard Thomas
Seconded: Jim Bryan, Frances Kirwan
Daniel Greb
Essener Seminar für Algebraische Geometrie und Arithmetik
Fakultät für Mathematik
Universität Duisburg-Essen
D-45117 Essen
Julius Ross
Department of Pure Mathematics and Mathematical Statistics
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
Matei Toma
Institut de Mathématiques Élie Cartan
Université de Lorraine
BP 70239
54506 Vandoeuvre-lès-Nancy Cedex