Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 233-261.

Using an existence criterion for good moduli spaces of Artin stacks by Alper–Fedorchuk–Smyth we construct a proper moduli space of rank two sheaves with fixed Chern classes on a given complex projective manifold that are Gieseker-Maruyama-semistable with respect to a fixed Kähler class.

En utilisant le critère d’existence d’un bon espace de modules d’un champ d’Artin dû à Alper–Fedorchuk–Smyth, nous construisons un espace de modules propre de faisceaux de rang 2 sur une variété projective complexe donnée, de classes de Chern fixées et qui sont Gieseker-Maruyama-semistables par rapport à une classe de Kähler fixée.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.116
Classification: 32G13,  14D20,  14D23,  14J60
Keywords: Kähler manifolds, moduli of coherent sheaves, algebraic stacks, good moduli spaces, semi-universal deformations, local quotient presentations
Daniel Greb 1; Matei Toma 2

1 Essener Seminar für Algebraische Geometrie und Arithmetik, Fakultät für Mathematik, Universität Duisburg–Essen 45112 Essen, Germany
2 Université de Lorraine, CNRS, IECL F-54000 Nancy, France
@article{JEP_2020__7__233_0,
     author = {Daniel Greb and Matei Toma},
     title = {Moduli spaces of sheaves that are semistable with respect to a {K\"ahler} polarisation},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {233--261},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.116},
     zbl = {07152736},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.116/}
}
TY  - JOUR
TI  - Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2020
DA  - 2020///
SP  - 233
EP  - 261
VL  - 7
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.116/
UR  - https://zbmath.org/?q=an%3A07152736
UR  - https://doi.org/10.5802/jep.116
DO  - 10.5802/jep.116
LA  - en
ID  - JEP_2020__7__233_0
ER  - 
%0 Journal Article
%T Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation
%J Journal de l’École polytechnique — Mathématiques
%D 2020
%P 233-261
%V 7
%I École polytechnique
%U https://doi.org/10.5802/jep.116
%R 10.5802/jep.116
%G en
%F JEP_2020__7__233_0
Daniel Greb; Matei Toma. Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 233-261. doi : 10.5802/jep.116. https://jep.centre-mersenne.org/articles/10.5802/jep.116/

[AFS17] J. Alper, M. Fedorchuk & D. I. Smyth - “Second flip in the Hassett-Keel program: existence of good moduli spaces”, Compositio Math. 153 (2017) no. 8, p. 1584-1609 | Article | MR: 3649808 | Zbl: 1403.14038

[AHLH18] J. Alper, D. Halper-Leistner & J. Heinloth - “Existence of moduli spaces for algebraic stacks”, 2018 | 1812.01128

[AHR15] J. Alper, J. Hall & D. Rydh - “A Luna étale slice theorem for algebraic stacks”, 2015 | 1504.06467

[AK16] J. Alper & A. Kresch - “Equivariant versal deformations of semistable curves”, Michigan Math. J. 65 (2016) no. 2, p. 227-250 | Article | MR: 3510906 | Zbl: 1346.14069

[Alp13] J. Alper - “Good moduli spaces for Artin stacks”, Ann. Inst. Fourier (Grenoble) 63 (2013) no. 6, p. 2349-2402 | Article | Numdam | MR: 3237451 | Zbl: 1314.14095

[Alp15] J. Alper - “Artin algebraization and quotient stacks”, 2015 | 1510.07804

[Ant19] S. Antonakoudis - “Criteria of separatedness and properness”, 2019, notes available at https://www.dpmms.cam.ac.uk/~sa443/papers/criteria.pdf

[BTT17] N. Buchdahl, A. Teleman & M. Toma - “A continuity theorem for families of sheaves on complex surfaces”, J. Topology 10 (2017) no. 4, p. 995-1028 | Article | MR: 3743066 | Zbl: 1394.32010

[Dré04] J.-M. Drézet - “Luna’s slice theorem and applications”, in Algebraic group actions and quotients, Hindawi Publ. Corp., Cairo, 2004, p. 39-89 | Zbl: 1109.14307

[Eis95] D. Eisenbud - Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Math., vol. 150, Springer-Verlag, New York, 1995 | Article | Zbl: 0819.13001

[Fle78] H. Flenner - Deformationen holomorpher Abbildungen, Osnabrücker Schriften zur Mathematik (Reihe P), vol. 8, Fachbereich Mathematik, Univ. Osnabrük, 1978, available online at http://www.ruhr-uni-bochum.de/imperia/md/content/mathematik/lehrstuhli/deformationen.pdf

[Gie77] D. Gieseker - “On the moduli of vector bundles on an algebraic surface”, Ann. of Math. (2) 106 (1977) no. 1, p. 45-60 | Article | MR: 466475 | Zbl: 0381.14003

[GRT16a] D. Greb, J. Ross & M. Toma - “Moduli of vector bundles on higher-dimensional base manifolds—construction and variation”, Internat. J. Math. 27 (2016) no. 7, article ID 1650054, 27 pages | Article | MR: 3605660 | Zbl: 06617518

[GRT16b] D. Greb, J. Ross & M. Toma - “Variation of Gieseker moduli spaces via quiver GIT”, Geom. Topol. 20 (2016) no. 3, p. 1539-1610 | Article | MR: 3523063 | Zbl: 1400.14032

[GT17] D. Greb & M. Toma - “Compact moduli spaces for slope-semistable sheaves”, Algebraic Geom. 4 (2017) no. 1, p. 40-78 | Article | MR: 3592465 | Zbl: 1423.14076

[Har77] R. Hartshorne - Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, New York-Heidelberg, 1977 | Zbl: 0367.14001

[Har10] R. Hartshorne - Deformation theory, Graduate Texts in Math., vol. 257, Springer, New York, 2010 | Article | MR: 2583634 | Zbl: 1186.14004

[HL10] D. Huybrechts & M. Lehn - The geometry of moduli spaces of sheaves, Cambridge Math. Library, Cambridge University Press, Cambridge, 2010 | Article | Zbl: 1206.14027

[HMP98] P. Heinzner, L. Migliorini & M. Polito - “Semistable quotients”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998) no. 2, p. 233-248 | Numdam | MR: 1631577 | Zbl: 0922.32017

[JS12] D. Joyce & Y. Song - A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc., vol. 217, no. 1020, American Mathematical Society, Providence, RI, 2012 | Article | Zbl: 1259.14054

[Kem93] G. R. Kempf - Algebraic varieties, London Math. Society Lect. Note Ser., vol. 172, Cambridge University Press, Cambridge, 1993 | Article | MR: 1252397 | Zbl: 0780.14001

[Knu71] D. Knutson - Algebraic spaces, Lect. Notes in Math., vol. 203, Springer-Verlag, Berlin-New York, 1971 | MR: 302647 | Zbl: 0221.14001

[KS90] S. Kosarew & H. Stieber - “A construction of maximal modular subspaces in local deformation theory”, Abh. Math. Sem. Univ. Hamburg 60 (1990), p. 17-36 | Article | MR: 1087115 | Zbl: 0733.32017

[Lan75] S. G. Langton - “Valuative criteria for families of vector bundles on algebraic varieties”, Ann. of Math. (2) 101 (1975), p. 88-110 | Article | MR: 364255 | Zbl: 0307.14007

[Lan83] H. Lange - “Universal families of extensions”, J. Algebra 83 (1983) no. 1, p. 101-112 | Article | MR: 710589 | Zbl: 0518.14008

[LMB00] G. Laumon & L. Moret-Bailly - Champs algébriques, Ergeb. Math. Grenzgeb. (3), vol. 39, Springer-Verlag, Berlin, 2000 | Zbl: 0945.14005

[LP97] J. Le Potier - Lectures on vector bundles, Cambridge Studies in Advanced Math., vol. 54, Cambridge University Press, Cambridge, 1997 | MR: 1428426 | Zbl: 0872.14003

[Pal90] V. P. Palamodov - “Deformations of complex spaces”, in Several complex variables. IV. Algebraic aspects of complex analysis (G. M. Khenkin, ed.), Encycl. Math. Sci., vol. 10, Springer-Verlag, Berlin, 1990, p. 105-194

[Ses67] C. S. Seshadri - “Space of unitary vector bundles on a compact Riemann surface”, Ann. of Math. (2) 85 (1967), p. 303-336 | Article | MR: 233371 | Zbl: 0173.23001

[Sta19] Stacks Project Authors - “The Stacks Project”, 2019, https://stacks.math.columbia.edu

[Tel08] A. Teleman - “Families of holomorphic bundles”, Commun. Contemp. Math. 10 (2008) no. 4, p. 523-551 | Article | MR: 2444847 | Zbl: 1159.32011

[Tom16] M. Toma - “Bounded sets of sheaves on Kähler manifolds”, J. reine angew. Math. 710 (2016), p. 77-93 | Article | Zbl: 1342.32010

[Tom17] M. Toma - “Properness criteria for families of coherent analytic sheaves”, 2017, to appear in Algebraic Geom. | 1710.01484

[Tom19] M. Toma - “Bounded sets of sheaves on Kähler manifolds. II”, 2019 | 1906.05853

Cited by Sources: