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 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.
Accepté le :
Publié le :
DOI : 10.5802/jep.116
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
@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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.5802/jep.116/
[AFS17] 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] Existence of moduli spaces for algebraic stacks, 2018 | arXiv
[AHR15] A Luna étale slice theorem for algebraic stacks, 2015 | arXiv
[AK16] Equivariant versal deformations of semistable curves, Michigan Math. J., Volume 65 (2016) no. 2, pp. 227-250 | DOI | MR | Zbl
[Alp13] 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] Artin algebraization and quotient stacks, 2015 | arXiv
[Ant19] Criteria of separatedness and properness, 2019 (notes available at https://www.dpmms.cam.ac.uk/~sa443/papers/criteria.pdf)
[BTT17] A continuity theorem for families of sheaves on complex surfaces, J. Topology, Volume 10 (2017) no. 4, pp. 995-1028 | DOI | MR | Zbl
[Dré04] Luna’s slice theorem and applications, Algebraic group actions and quotients, Hindawi Publ. Corp., Cairo, 2004, pp. 39-89 | Zbl
[Eis95] Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Math., 150, Springer-Verlag, New York, 1995 | DOI | Zbl
[Fle78] 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] 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] 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] Variation of Gieseker moduli spaces via quiver GIT, Geom. Topol., Volume 20 (2016) no. 3, pp. 1539-1610 | DOI | MR | Zbl
[GT17] Compact moduli spaces for slope-semistable sheaves, Algebraic Geom., Volume 4 (2017) no. 1, pp. 40-78 | DOI | MR | Zbl
[Har77] Algebraic geometry, Graduate Texts in Math., 52, Springer-Verlag, New York-Heidelberg, 1977 | Zbl
[Har10] Deformation theory, Graduate Texts in Math., 257, Springer, New York, 2010 | DOI | MR | Zbl
[HL10] The geometry of moduli spaces of sheaves, Cambridge Math. Library, Cambridge University Press, Cambridge, 2010 | DOI | Zbl
[HMP98] Semistable quotients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 26 (1998) no. 2, pp. 233-248 | Numdam | MR | Zbl
[JS12] A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc., 217, no. 1020, American Mathematical Society, Providence, RI, 2012 | DOI | Zbl
[Kem93] Algebraic varieties, London Math. Society Lect. Note Ser., 172, Cambridge University Press, Cambridge, 1993 | DOI | MR | Zbl
[Knu71] Algebraic spaces, Lect. Notes in Math., 203, Springer-Verlag, Berlin-New York, 1971, vi+261 pages | MR | Zbl
[KS90] 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] Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. (2), Volume 101 (1975), pp. 88-110 | DOI | MR | Zbl
[Lan83] Universal families of extensions, J. Algebra, Volume 83 (1983) no. 1, pp. 101-112 | DOI | MR | Zbl
[LMB00] Champs algébriques, Ergeb. Math. Grenzgeb. (3), 39, Springer-Verlag, Berlin, 2000 | Zbl
[LP97] Lectures on vector bundles, Cambridge Studies in Advanced Math., 54, Cambridge University Press, Cambridge, 1997 | MR | Zbl
[Pal90] 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] Space of unitary vector bundles on a compact Riemann surface, Ann. of Math. (2), Volume 85 (1967), pp. 303-336 | DOI | MR | Zbl
[Sta19] The Stacks Project, 2019 (https://stacks.math.columbia.edu)
[Tel08] Families of holomorphic bundles, Commun. Contemp. Math., Volume 10 (2008) no. 4, pp. 523-551 | DOI | MR | Zbl
[Tom16] Bounded sets of sheaves on Kähler manifolds, J. reine angew. Math., Volume 710 (2016), pp. 77-93 | DOI | Zbl
[Tom17] Properness criteria for families of coherent analytic sheaves, 2017 (to appear in Algebraic Geom.) | arXiv
[Tom19] Bounded sets of sheaves on Kähler manifolds. II, 2019 | arXiv
Cité par Sources :