Variation of Gieseker moduli spaces via quiver GIT

Greb, Daniel¹ ; Ross, Julius² ; Toma, Matei³

¹ Essener Seminar für Algebraische Geometrie und Arithmetik, Fakultät für Mathematik, Universität Duisburg-Essen, D-45117 Essen, Germany
² Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK
³ Institut de Mathématiques Élie Cartan, Université de Lorraine, BP 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France

Geometry & topology, Tome 20 (2016) no. 3, pp. 1539-1610.

Résumé

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.

Détail

DOI : 10.2140/gt.2016.20.1539

Classification : 14D20, 14J60, 32G13, 14L24, 16G20
Keywords: Gieseker stability, variation of moduli spaces, chamber structures, boundedness, moduli of quiver representations, semistable sheaves on Kähler manifolds

Bibliographie

[1] L Álvarez-Cónsul, Some results on the moduli spaces of quiver bundles, Geom. Dedicata 139 (2009) 99

[2] L Álvarez-Cónsul, A King, A functorial construction of moduli of sheaves, Invent. Math. 168 (2007) 613

[3] L Álvarez-Cónsul, A King, Moduli of sheaves from moduli of Kronecker modules, from: "Moduli spaces and vector bundles" (editors S B Bradlow, L Brambila-Paz, O García-Prada, S Ramanan), London Math. Soc. Lecture Note Ser. 359, Cambridge Univ. Press (2009) 212

[4] I V Arzhantsev, J Hausen, Geometric invariant theory via Cox rings, J. Pure Appl. Algebra 213 (2009) 154

[5] A Bertram, Some remarks on surface moduli and determinants, from: "Recent advances in algebraic geometry" (editors C D Hacon, M Mustaţă, M Popa), London Math. Soc. Lecture Note Ser. 417, Cambridge Univ. Press (2015) 13

[6] N Bourbaki, Éléments de mathématique : Algèbre, Chapitres 1 à 3, Hermann (1970)

[7] C Chindris, Notes on GIT-fans for quivers, preprint (2008)

[8] O Debarre, Higher-dimensional algebraic geometry, Springer (2001)

[9] I V Dolgachev, Y Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. (1998) 5

[10] S K Donaldson, Irrationality and the h–cobordism conjecture, J. Differential Geom. 26 (1987) 141

[11] G Ellingsrud, L Göttsche, Variation of moduli spaces and Donaldson invariants under change of polarization, J. Reine Angew. Math. 467 (1995) 1

[12] R Friedman, J W Morgan, On the diffeomorphism types of certain algebraic surfaces, I, J. Differential Geom. 27 (1988) 297

[13] R Friedman, Z Qin, Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces, Comm. Anal. Geom. 3 (1995) 11

[14] W Fulton, Intersection theory, 2, Springer (1998)

[15] V Ginzburg, Lectures on Nakajima’s quiver varieties, from: "Geometric methods in representation theory, I" (editor M Brion), Sémin. Congr. 24, Soc. Math. France (2012) 145

[16] L Göttsche, Change of polarization and Hodge numbers of moduli spaces of torsion free sheaves on surfaces, Math. Z. 223 (1996) 247

[17] D Greb, Projectivity of analytic Hilbert and Kähler quotients, Trans. Amer. Math. Soc. 362 (2010) 3243

[18] D Greb, J Ross, M Toma, Semi-continuity of stability for sheaves and variation of Gieseker moduli spaces, preprint (2015)

[19] D Greb, M Toma, Compact moduli spaces for slope-semistable sheaves, preprint (2013)

[20] 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 (1966)

[21] M Halic, Quotients of affine spaces for actions of reductive groups, preprint (2004)

[22] R Hartshorne, Algebraic geometry, 52, Springer (1977)

[23] P Heinzner, L Migliorini, Projectivity of moment map quotients, Osaka J. Math. 38 (2001) 167

[24] Y Hu, W P Li, Variation of the Gieseker and Uhlenbeck compactifications, Internat. J. Math. 6 (1995) 397

[25] D Huybrechts, M Lehn, The geometry of moduli spaces of sheaves, Cambridge Univ. Press (2010)

[26] D Joyce, Configurations in abelian categories, I : Basic properties and moduli stacks, Adv. Math. 203 (2006) 194

[27] D Joyce, Configurations in abelian categories, II : Ringel–Hall algebras, Adv. Math. 210 (2007) 635

[28] D Joyce, Configurations in abelian categories, III : Stability conditions and identities, Adv. Math. 215 (2007) 153

[29] D Joyce, Configurations in abelian categories, IV : Invariants and changing stability conditions, Adv. Math. 217 (2008) 125

[30] D Joyce, Y Song, A theory of generalized Donaldson–Thomas invariants, 1020, Amer. Math. Soc. (2012)

[31] Y H Kiem, J Li, A wall crossing formula of Donaldson–Thomas invariants without Chern–Simons functional, Asian J. Math. 17 (2013) 63

[32] A D King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. 45 (1994) 515

[33] A Langer, Semistable sheaves in positive characteristic, Ann. of Math. 159 (2004) 251

[34] S G Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. 101 (1975) 88

[35] A Laudin, A Schmitt, Recent results on quiver sheaves, Cent. Eur. J. Math. 10 (2012) 1246

[36] J Le Potier, L’espace de modules de Simpson, exposé, SGA Jussieu (1992)

[37] S Mac Lane, Categories for the working mathematician, 5, Springer (1998)

[38] K Matsuki, R Wentworth, Mumford–Thaddeus principle on the moduli space of vector bundles on an algebraic surface, Internat. J. Math. 8 (1997) 97

[39] D Mumford, J Fogarty, F Kirwan, Geometric invariant theory, 34, Springer (1994)

[40] Z Qin, Equivalence classes of polarizations and moduli spaces of sheaves, J. Differential Geom. 37 (1993) 397

[41] A Rudakov, Stability for an abelian category, J. Algebra 197 (1997) 231

[42] 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

[43] A Schmitt, Moduli for decorated tuples of sheaves and representation spaces for quivers, Proc. Indian Acad. Sci. Math. Sci. 115 (2005) 15

[44] A H W Schmitt, Geometric invariant theory and decorated principal bundles, European Math. Soc. (2008)

[45] C T Simpson, Moduli of representations of the fundamental group of a smooth projective variety, I, Inst. Hautes Études Sci. Publ. Math. (1994) 47

[46] A Teleman, Families of holomorphic bundles, Commun. Contemp. Math. 10 (2008) 523

[47] M Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996) 691

[48] 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

[49] K Yamada, Blowing-ups describing the polarization change of moduli schemes of semistable sheaves of general rank, Comm. Algebra 38 (2010) 3094