Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation

Greb, Daniel; Toma, Matei

doi:10.5802/jep.116

Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation
[Espaces de modules de faisceaux semistables par rapport à une polarisation kählérienne]

Greb, Daniel ¹ ; Toma, Matei 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

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 233-261.

Résumé
Abstract

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.

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.

Reçu le : 2018-08-15
Accepté le : 2019-11-25
Publié le : 2020-01-15

Zbl

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
Mot clés : Variétés de Kähler, modules de faisceaux cohérents, champs algébriques, bons espaces de modules, déformations semi-universelles, présentations quotient locales

Affiliations des auteurs :

Greb, Daniel ¹ ; Toma, Matei ²

@article{JEP_2020__7__233_0,
     author = {Greb, Daniel and Toma, Matei},
     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 = {http://www.numdam.org/articles/10.5802/jep.116/}
}

TY  - JOUR
AU  - Greb, Daniel
AU  - Toma, Matei
TI  - Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2020
SP  - 233
EP  - 261
VL  - 7
PB  - Ecole polytechnique
UR  - http://www.numdam.org/articles/10.5802/jep.116/
DO  - 10.5802/jep.116
LA  - en
ID  - JEP_2020__7__233_0
ER  -

%0 Journal Article
%A Greb, Daniel
%A Toma, Matei
%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 Ecole polytechnique
%U http://www.numdam.org/articles/10.5802/jep.116/
%R 10.5802/jep.116
%G en
%F JEP_2020__7__233_0

Greb, Daniel; Toma, Matei. Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 233-261. doi : 10.5802/jep.116. http://www.numdam.org/articles/10.5802/jep.116/

Bibliographie
Cité par

[AFS17] Alper, Jarod; Fedorchuk, Maksym; Smyth, David Ishii Second flip in the Hassett-Keel program: existence of good moduli spaces, Compositio Math., Volume 153 (2017) no. 8, pp. 1584-1609 | DOI | MR | Zbl

[AHLH18] Alper, Jarod; Halper-Leistner, Daniel; Heinloth, Jochen Existence of moduli spaces for algebraic stacks, 2018 | arXiv

[AHR15] Alper, Jarod; Hall, Jack; Rydh, David A Luna étale slice theorem for algebraic stacks, 2015 | arXiv

[AK16] Alper, Jarod; Kresch, Andrew Equivariant versal deformations of semistable curves, Michigan Math. J., Volume 65 (2016) no. 2, pp. 227-250 | DOI | MR | Zbl

[Alp13] Alper, Jarod Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble), Volume 63 (2013) no. 6, pp. 2349-2402 http://aif.cedram.org/item?id=AIF_2013__63_6_2349_0 | DOI | Numdam | MR | Zbl

[Alp15] Alper, Jarod Artin algebraization and quotient stacks, 2015 | arXiv

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

[BTT17] Buchdahl, Nicholas; Teleman, Andrei; Toma, Matei A continuity theorem for families of sheaves on complex surfaces, J. Topology, Volume 10 (2017) no. 4, pp. 995-1028 | DOI | MR | Zbl

[Dré04] Drézet, Jean-Marc Luna’s slice theorem and applications, Algebraic group actions and quotients, Hindawi Publ. Corp., Cairo, 2004, pp. 39-89 | Zbl

[Eis95] Eisenbud, David Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Math., 150, Springer-Verlag, New York, 1995 | DOI | Zbl

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

[Gie77] Gieseker, D. On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2), Volume 106 (1977) no. 1, pp. 45-60 | DOI | MR | Zbl

[GRT16a] Greb, Daniel; Ross, Julius; Toma, Matei Moduli of vector bundles on higher-dimensional base manifolds—construction and variation, Internat. J. Math., Volume 27 (2016) no. 7, 1650054, 27 pages | DOI | MR | Zbl

[GRT16b] Greb, Daniel; Ross, Julius; Toma, Matei Variation of Gieseker moduli spaces via quiver GIT, Geom. Topol., Volume 20 (2016) no. 3, pp. 1539-1610 | DOI | MR | Zbl

[GT17] Greb, Daniel; Toma, Matei Compact moduli spaces for slope-semistable sheaves, Algebraic Geom., Volume 4 (2017) no. 1, pp. 40-78 | DOI | MR | Zbl

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

[Har10] Hartshorne, Robin Deformation theory, Graduate Texts in Math., 257, Springer, New York, 2010 | DOI | MR | Zbl

[HL10] Huybrechts, Daniel; Lehn, Manfred The geometry of moduli spaces of sheaves, Cambridge Math. Library, Cambridge University Press, Cambridge, 2010 | DOI | Zbl

[HMP98] Heinzner, Peter; Migliorini, Luca; Polito, Marzia Semistable quotients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 26 (1998) no. 2, pp. 233-248 | Numdam | MR | Zbl

[JS12] Joyce, Dominic; Song, Yinan A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc., 217, no. 1020, American Mathematical Society, Providence, RI, 2012 | DOI | Zbl

[Kem93] Kempf, George R. Algebraic varieties, London Math. Society Lect. Note Ser., 172, Cambridge University Press, Cambridge, 1993 | DOI | MR | Zbl

[Knu71] Knutson, Donald Algebraic spaces, Lect. Notes in Math., 203, Springer-Verlag, Berlin-New York, 1971, vi+261 pages | MR | Zbl

[KS90] Kosarew, S.; Stieber, H. A construction of maximal modular subspaces in local deformation theory, Abh. Math. Sem. Univ. Hamburg, Volume 60 (1990), pp. 17-36 | DOI | MR | Zbl

[Lan75] Langton, Stacy G. Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. (2), Volume 101 (1975), pp. 88-110 | DOI | MR | Zbl

[Lan83] Lange, Herbert Universal families of extensions, J. Algebra, Volume 83 (1983) no. 1, pp. 101-112 | DOI | MR | Zbl

[LMB00] Laumon, Gérard; Moret-Bailly, Laurent Champs algébriques, Ergeb. Math. Grenzgeb. (3), 39, Springer-Verlag, Berlin, 2000 | Zbl

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

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

[Ses67] Seshadri, C. S. Space of unitary vector bundles on a compact Riemann surface, Ann. of Math. (2), Volume 85 (1967), pp. 303-336 | DOI | MR | Zbl

[Sta19] Stacks Project Authors The Stacks Project, 2019 (https://stacks.math.columbia.edu)

[Tel08] Teleman, Andrei Families of holomorphic bundles, Commun. Contemp. Math., Volume 10 (2008) no. 4, pp. 523-551 | DOI | MR | Zbl

[Tom16] Toma, Matei Bounded sets of sheaves on Kähler manifolds, J. reine angew. Math., Volume 710 (2016), pp. 77-93 | DOI | Zbl

[Tom17] Toma, Matei Properness criteria for families of coherent analytic sheaves, 2017 (to appear in Algebraic Geom.) | arXiv

[Tom19] Toma, Matei Bounded sets of sheaves on Kähler manifolds. II, 2019 | arXiv

Cité par Sources :