Hostname: page-component-cb9f654ff-p5m67 Total loading time: 0 Render date: 2025-08-31T06:38:44.822Z Has data issue: false hasContentIssue false
  • English
  • Français

MODULI SPACES OF SLOPE-SEMISTABLE SHEAVES WITH REFLEXIVE SESHADRI GRADUATIONS

Part of: Families, fibrations Deformations of analytic structures

Published online by Cambridge University Press:  22 August 2025

MIHAI PAVEL Open the ORCID record for MIHAI PAVEL [Opens in a new window]  and
MATEI TOMA Open the ORCID record for MATEI TOMA [Opens in a new window]
MIHAI PAVEL
Affiliation:
Institute of Mathematics of the Romanian Academy Bucharest 014700 Romania cpavel@imar.ro
MATEI TOMA*
Affiliation:
Institut Elie Cartan de Lorraine https://ror.org/04rvw8791 Université de Lorraine , CNRS, IECL Nancy F-54000 France
*
matei.toma@univ-lorraine.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the moduli stacks of slope-semistable torsion-free coherent sheaves that admit reflexive, respectively, locally free, Seshadri graduations on a smooth projective variety. We show that they are open in the stack of coherent sheaves and that they admit good moduli spaces when the field characteristic is zero. In addition, in the locally free case we prove that the resulting moduli space is a quasi-projective scheme.

Keywords

semistable coherent sheavesmoduli spaces

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles
Secondary: 32G13: Analytic moduli problems

Information

Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 19
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alper, J., Good moduli spaces for Artin stacks , Ann. Inst. Fourier 63 (2013), 2349–2402.10.5802/aif.2833CrossRefGoogle Scholar
Alper, J., Belmans, P., Bragg, D., Liang, J. and Tajakka, T., Projectivity of the Moduli Space of Vector Bundles on a Curve, Stacks project expository collection (SPEC), Cambridge University Press, Cambridge, 2022, 90–125.Google Scholar
Alper, J., Halpern-Leistner, D. and Heinloth, J., Existence of moduli spaces for algebraic stacks , Invent. Math. 234 (2023), 949–1038.10.1007/s00222-023-01214-4CrossRefGoogle Scholar
Buchdahl, N. and Schumacher, G., An analytic application of geometric invariant theory. II: Coarse moduli spaces , J. Geom. Phys. 175 (2022), Article ID 104467.10.1016/j.geomphys.2022.104467CrossRefGoogle Scholar
Buchdahl, N., Teleman, A. and Toma, M., A continuity theorem for families of sheaves on complex surfaces , J. Topol. 10 (2017), 995–1028.10.1112/topo.12029CrossRefGoogle Scholar
Campana, F. and Păun, M., Foliations with positive slopes and birational stability of orbifold cotangent bundles , Publ. Math., Inst. Hautes Étud. Sci. 129 (2019), 1–49.10.1007/s10240-019-00105-wCrossRefGoogle Scholar
Gieseker, D., On the moduli of vector bundles on an algebraic surface , Ann. Math. (2) 106 (1977), 45–60.10.2307/1971157CrossRefGoogle Scholar
Greb, D., Ross, J. and Toma, M., Moduli of vector bundles on higher-dimensional base manifolds—Construction and variation , Int. J. Math. 27 (2016), Article ID 1650054.10.1142/S0129167X16500543CrossRefGoogle Scholar
Greb, D., Sibley, B., Toma, M. and Wentworth, R., Complex algebraic compactifications of the moduli space of Hermitian Yang-Mills connections on a projective manifold , Geom. Topol. 25 (2021), 1719–1818.10.2140/gt.2021.25.1719CrossRefGoogle Scholar
Greb, D. and Toma, M., Compact moduli spaces for slope-semistable sheaves , Algebr. Geom. 4 (2017), 40–78.10.14231/AG-2017-003CrossRefGoogle Scholar
Greb, D. and Toma, M., Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation , J. Éc. Polytech., Math. 7 (2020), 233–261.10.5802/jep.116CrossRefGoogle Scholar
Hartshorne, R., Stable reflexive sheaves , Math. Ann. 254 (1980), 121–176.10.1007/BF01467074CrossRefGoogle Scholar
Huybrechts, D., Lectures on $K$ 3 Surfaces, Cambridge Studies in Advanced Mathematics. 158, Cambridge University Press, Cambridge, 2016, 121–176.Google Scholar
Huybrechts, D. and Lehn, M., The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge University Press, Cambridge, 2010.10.1017/CBO9780511711985CrossRefGoogle Scholar
Langer, A., Semistable sheaves in positive characteristic , Ann. Math. (2) 159 (2004), 251–276.10.4007/annals.2004.159.251CrossRefGoogle Scholar
Langton, S. G., Valuative criteria for families of vector bundles on algebraic varieties , Ann. Math. (2) 101 (1975), 88–110.10.2307/1970987CrossRefGoogle Scholar
Lieblich, M., Moduli of complexes on a proper morphism , J. Algebr. Geom. 15 (2006), 175–206.10.1090/S1056-3911-05-00418-2CrossRefGoogle Scholar
Maruyama, M., Moduli of stable sheaves. II , J. Math. Kyoto Univ. 18 (1978), 557–614.Google Scholar
Maruyama, M., On boundedness of families of torsion free sheaves , J. Math. Kyoto Univ. 21 (1981), 673–701.Google Scholar
Maruyama, M., Moduli Spaces of Stable Sheaves on Schemes: Restriction Theorems, Boundedness and the GIT Construction. With Collaboration of T. Abe and M. Inaba, Journal of Mathematics of Kyoto University, 33, Mathematical Society of Japan, Tokyo, 2016.10.2969/msjmemoirs/033010000CrossRefGoogle Scholar
Mayrand, M., Kempf-Ness type theorems and Nahm equations , J. Geom. Phys. 136 (2019), 138–155.10.1016/j.geomphys.2018.11.003CrossRefGoogle Scholar
Miyaoka, Y., “ The Chern classes and Kodaira dimension of a minimal variety ” in Oda, T. (ed.), Algebraic geometry, Proceeding of Symposium Sendai/Japan 1985, Advances in Pure and Applied Mathematics, 10, 1987, 449–476.Google Scholar
Mumford, D., “ Projective invariants of projective structures and applications ” in Proceedings of the International Congress of Mathematicians 1962, 1963, 526–530.Google Scholar
Mumford, D., Fogarty, J. and Kirwan, F. Geometric invariant Theory. 3rd ed., Vol. 34, Springer-Verlag, Berlin, 1993.Google Scholar
Narasimhan, M. S. and Seshadri, C. S., Stable and unitary vector bundles on a compact Riemann surface , Ann. Math. (2) 82 (1965), 540–567.10.2307/1970710CrossRefGoogle Scholar
Okonek, C., Schneider, M. and Spindler, H., Vector Bundles on Complex Projective Spaces, corrected reprint of the 1988 edition, with an appendix by S. I. Gelfand, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2011.Google Scholar
Olsson, M. and Starr, J., Quot functors for Deligne-Mumford stacks , Commun. Algebra 31 (2003), 4069–4096.10.1081/AGB-120022454CrossRefGoogle Scholar
Seshadri, C. S., Space of unitary vector bundles on a compact Riemann surface , Ann. Math. (2) 85 (1967), 303–336.10.2307/1970444CrossRefGoogle Scholar
Simpson, C. T., Moduli of representations of the fundamental group of a smooth projective variety, I . Publ. Math., Inst. Hautes Étud. Sci. 79 (1994), 47–129.10.1007/BF02698887CrossRefGoogle Scholar
Stacks project authors, The stacks project, 2020. https://stacks.math.columbia.edu Google Scholar
Tajakka, T., Uhlenbeck compactification as a Bridgeland moduli space , Int. Math. Res. Not. 2023 (2023), 4952–4997.10.1093/imrn/rnab372CrossRefGoogle Scholar
Takemoto, F., Stable vector bundles on algebraic surfaces , Nagoya Math. J. 47 (1972), 29–48.10.1017/S0027763000014896CrossRefGoogle Scholar
Toma, M., Holomorphe Vektorbündel auf Nichtalgebraischen Flächen, University of Bayreuth, Bayreuth, 1992. https://dev-iecl.univ-lorraine.fr/wp-content/uploads/2020/12/these.pdf.Google Scholar
Toma, M., Properness criteria for families of coherent analytic sheaves , Algebr. Geom. 7 (2020), 486–502.10.14231/AG-2020-015CrossRefGoogle Scholar
Toma, M., Bounded sets of sheaves on relative analytic spaces , Ann. Henri Lebesgue 4 (2021), 1531–1563.10.5802/ahl.109CrossRefGoogle Scholar
Weissmann, D. and Zhang, X., A stacky approach to identify the semi-stable locus of vector bundles , Algebr. Geom. 12 (2025), 262–298.10.14231/AG-2025-008CrossRefGoogle Scholar