Geometry & Topology

Variation of Gieseker moduli spaces via quiver GIT

Daniel Greb, Julius Ross, and Matei Toma

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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.

Article information

Geom. Topol., Volume 20, Number 3 (2016), 1539-1610.

Received: 26 September 2014
Revised: 5 June 2015
Accepted: 3 July 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] 32G13: Analytic moduli problems {For algebraic moduli problems, see 14D20, 14D22, 14H10, 14J10} [See also 14H15, 14J15]
Secondary: 14L24: Geometric invariant theory [See also 13A50] 16G20: Representations of quivers and partially ordered sets

Gieseker stability variation of moduli spaces chamber structures boundedness moduli of quiver representations semistable sheaves on Kähler manifolds


Greb, Daniel; Ross, Julius; Toma, Matei. Variation of Gieseker moduli spaces via quiver GIT. Geom. Topol. 20 (2016), no. 3, 1539--1610. doi:10.2140/gt.2016.20.1539.

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