#### 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.

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##### 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