## Geometry & Topology

### Variation of Gieseker moduli spaces via quiver GIT

#### 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 $ω ∈ N1(X)ℝ$ 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

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

Dates
Revised: 5 June 2015
Accepted: 3 July 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858997

Digital Object Identifier
doi:10.2140/gt.2016.20.1539

Mathematical Reviews number (MathSciNet)
MR3523063

Zentralblatt MATH identifier
06624253

#### Citation

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. https://projecteuclid.org/euclid.gt/1510858997

#### References

• L Álvarez-Cónsul, Some results on the moduli spaces of quiver bundles, Geom. Dedicata 139 (2009) 99–120
• L Álvarez-Cónsul, A King, A functorial construction of moduli of sheaves, Invent. Math. 168 (2007) 613–666
• L Álvarez-Cónsul, A King, Moduli of sheaves from moduli of Kronecker modules, from: “Moduli spaces and vector bundles”, (S B Bradlow, L Brambila-Paz, O García-Prada, S Ramanan, editors), London Math. Soc. Lecture Note Ser. 359, Cambridge Univ. Press (2009) 212–228
• I V Arzhantsev, J Hausen, Geometric invariant theory via Cox rings, J. Pure Appl. Algebra 213 (2009) 154–172
• A Bertram, Some remarks on surface moduli and determinants, from: “Recent advances in algebraic geometry”, (C D Hacon, M Mustaţă, M Popa, editors), London Math. Soc. Lecture Note Ser. 417, Cambridge Univ. Press (2015) 13–28
• N Bourbaki, Éléments de mathématique: Algèbre, Chapitres 1 à 3, Hermann, Paris (1970)
• C Chindris, Notes on GIT-fans for quivers, preprint (2008)
• O Debarre, Higher-dimensional algebraic geometry, Springer, New York (2001)
• I V Dolgachev, Y Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. (1998) 5–56
• S K Donaldson, Irrationality and the $h$–cobordism conjecture, J. Differential Geom. 26 (1987) 141–168
• G Ellingsrud, L G öttsche, Variation of moduli spaces and Donaldson invariants under change of polarization, J. Reine Angew. Math. 467 (1995) 1–49
• R Friedman, J W Morgan, On the diffeomorphism types of certain algebraic surfaces, I, J. Differential Geom. 27 (1988) 297–369
• R Friedman, Z Qin, Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces, Comm. Anal. Geom. 3 (1995) 11–83
• W Fulton, Intersection theory, 2nd edition, Ergeb. Math. Grenzgeb. 2, Springer, Berlin (1998)
• V Ginzburg, Lectures on Nakajima's quiver varieties, from: “Geometric methods in representation theory, I”, (M Brion, editor), Sémin. Congr. 24, Soc. Math. France, Paris (2012) 145–219
• L G öttsche, Change of polarization and Hodge numbers of moduli spaces of torsion free sheaves on surfaces, Math. Z. 223 (1996) 247–260
• D Greb, Projectivity of analytic Hilbert and Kähler quotients, Trans. Amer. Math. Soc. 362 (2010) 3243–3271
• D Greb, J Ross, M Toma, Semi-continuity of stability for sheaves and variation of Gieseker moduli spaces, preprint (2015) to appear in J. Reine Ang. Math.
• D Greb, M Toma, Compact moduli spaces for slope-semistable sheaves, preprint (2013) to appear in Alg. Geom.
• A Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique, IV: Les schémas de Hilbert, from: “Séminaire Bourbaki $1960/1961$ (Exposè 221)”, W A Benjamin, New York (1966) Reprinted as pages 249–276 in Séminaire Bourbaki 6, Soc. Math. France, Paris, 1995
• M Halic, Quotients of affine spaces for actions of reductive groups, preprint (2004)
• R Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York (1977)
• P Heinzner, L Migliorini, Projectivity of moment map quotients, Osaka J. Math. 38 (2001) 167–184
• Y Hu, W-P Li, Variation of the Gieseker and Uhlenbeck compactifications, Internat. J. Math. 6 (1995) 397–418
• D Huybrechts, M Lehn, The geometry of moduli spaces of sheaves, 2nd edition, Cambridge Univ. Press (2010)
• D Joyce, Configurations in abelian categories, I: Basic properties and moduli stacks, Adv. Math. 203 (2006) 194–255
• D Joyce, Configurations in abelian categories, II: Ringel–Hall algebras, Adv. Math. 210 (2007) 635–706
• D Joyce, Configurations in abelian categories, III: Stability conditions and identities, Adv. Math. 215 (2007) 153–219
• D Joyce, Configurations in abelian categories, IV: Invariants and changing stability conditions, Adv. Math. 217 (2008) 125–204
• D Joyce, Y Song, A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 1020, Amer. Math. Soc., Providence, RI (2012)
• Y-H Kiem, J Li, A wall crossing formula of Donaldson–Thomas invariants without Chern–Simons functional, Asian J. Math. 17 (2013) 63–94
• A D King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. 45 (1994) 515–530
• A Langer, Semistable sheaves in positive characteristic, Ann. of Math. 159 (2004) 251–276
• S G Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. 101 (1975) 88–110
• A Laudin, A Schmitt, Recent results on quiver sheaves, Cent. Eur. J. Math. 10 (2012) 1246–1279
• J Le Potier, L'espace de modules de Simpson, exposé, SGA Jussieu (1992) Available at \setbox0\makeatletter\@url https://www.imj-prg.fr/tga/jlp/scandrezetX1.pdf {\unhbox0
• S Mac Lane, Categories for the working mathematician, 2nd edition, Graduate Texts in Mathematics 5, Springer, New York (1998)
• K Matsuki, R Wentworth, Mumford–Thaddeus principle on the moduli space of vector bundles on an algebraic surface, Internat. J. Math. 8 (1997) 97–148
• D Mumford, J Fogarty, F Kirwan, Geometric invariant theory, 3rd edition, Ergeb. Math. Grenzgeb. 34, Springer, Berlin (1994)
• Z Qin, Equivalence classes of polarizations and moduli spaces of sheaves, J. Differential Geom. 37 (1993) 397–415
• A Rudakov, Stability for an abelian category, J. Algebra 197 (1997) 231–245
• A Schmitt, Walls for Gieseker semistability and the Mumford–Thaddeus principle for moduli spaces of sheaves over higher dimensional bases, Comment. Math. Helv. 75 (2000) 216–231
• A Schmitt, Moduli for decorated tuples of sheaves and representation spaces for quivers, Proc. Indian Acad. Sci. Math. Sci. 115 (2005) 15–49
• A H W Schmitt, Geometric invariant theory and decorated principal bundles, European Math. Soc., Zürich (2008)
• C T Simpson, Moduli of representations of the fundamental group of a smooth projective variety, I, Inst. Hautes Études Sci. Publ. Math. (1994) 47–129
• A Teleman, Families of holomorphic bundles, Commun. Contemp. Math. 10 (2008) 523–551
• M Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996) 691–723
• K Yamada, A sequence of blowing-ups connecting moduli of sheaves and the Donaldson polynomial under change of polarization, J. Math. Kyoto Univ. 43 (2004) 829–878
• K Yamada, Blowing-ups describing the polarization change of moduli schemes of semistable sheaves of general rank, Comm. Algebra 38 (2010) 3094–3110