Skip to main content Accessibility help
×
Home

Compactifications of reductive groups as moduli stacks of bundles

  • Johan Martens (a1) and Michael Thaddeus (a2)

Abstract

Let $G$ be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal $G$ -bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of $G$ . All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple $G$ , which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev–Manin’s spaces of weighted pointed curves and with Kausz’s compactification of $GL_{n}$ .

Copyright

References

Hide All
[ACV03]Abramovich, D., Corti, A. and Vistoli, A., Twisted bundles and admissible covers, Comm. Algebra 31 (2003), 35473618.
[AOV08]Abramovich, D., Olsson, M. and Vistoli, A., Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008), 10571091.
[AB04]Alexeev, V. and Brion, M., Stable reductive varieties I: affine varieties, Invent. Math. 157 (2004), 227274.
[Art69]Artin, M., Algebraization of formal moduli: I, in Global analysis: papers in honor of K. Kodaira, eds Spencer, D. C. and Iyanaga, S. (University of Tokyo and Princeton University Press, 1969), 2171.
[Art74]Artin, M., Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165189.
[BB11]Batyrev, V. and Blume, M., On generalisations of Losev–Manin moduli spaces for classical root systems, Pure Appl. Math. Q. 7 (2011), 10531084; doi: 10.4310/PAMQ.2011.v7.n4.a2.
[BN09]Biswas, I. and Nagaraj, D. S., Principal bundles over the projective line, J. Algebra 322 (2009), 34783491.
[BCS05]Borisov, L. A., Chen, L. and Smith, G. G., The orbifold Chow ring of toric Deligne–Mumford stacks, J. Amer. Math. Soc. 18 (2005), 193215.
[Bri07]Brion, M., The total coordinate ring of a wonderful variety, J. Algebra 313 (2007), 6199.
[Bri11]Brion, M., On automorphism groups of fiber bundles, Publ. Mat. Urug. 12 (2011), 3966, arXiv:1012.4606.
[Con05]Conrad, B., The Keel–Mori theorem via stacks, Preprint (2005),http://math.stanford.edu/∼conrad/papers/coarsespace.pdf.
[CGP10]Conrad, B., Gabber, O. and Prasad, G., Pseudo-reductive groups (Cambridge University Press, 2010).
[Cox95]Cox, D. A., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 1750.
[CLS11]Cox, D., Little, J. and Schenck, H., Toric varieties (American Mathematical Society, 2011).
[DCP82]De Concini, C. and Procesi, C., Complete symmetric varieties, in Invariant theory (Montecatini, 1982), Lecture Notes in Mathematics, vol. 996 (Springer, 1983), 144.
[DCP83]De Concini, C. and Procesi, C., Complete symmetric varieties II: intersection theory, in Algebraic groups and related topics (Kyoto/Nagoya, 1983), Advanced Studies in Pure Mathematics, vol. 6 (North-Holland, 1985), 481513.
[DM69]Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 75109.
[DG70]Demazure, M. and Gabriel, P., Groupes algébriques (North-Holland, 1970).
[Dol03]Dolgachev, I., Lectures on invariant theory, London Mathematical Society Lecture Note Series, vol. 296 (Cambridge, 2003).
[EGA]Grothendieck, A., Eléments de géométrie algébrique, Publ. Math. Inst. Hautes Études Sci. 4 (1960), [vol. I], 8 (1961) [vol. II], 11 (1961), 17 (1963) [vol. III], 20 (1964), 24 (1965), 28 (1966), 32 (1967) [vol. IV].
[EJ08]Evens, S. and Jones, B. F., On the wonderful compactification, Preprint (2008),arXiv:0801.0456.
[FMN10]Fantechi, B., Mann, E. and Nironi, F., Smooth toric Deligne–Mumford stacks, J. Reine Angew. Math. 648 (2010), 201244.
[Ful05]Fulghesu, D., On the Chow ring of the stack of rational nodal curves, PhD thesis, Scuola Normale Superiore di Pisa (2005), http://fulghesu.eu/doc/tesi_perfezionamento.pdf.
[Gie84]Gieseker, D., A degeneration of the moduli space of stable bundles, J. Differential Geom. 19 (1984), 173206.
[Gro57]Grothendieck, A., Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121138.
[GHH14]Gulbrandsen, M. G., Halle, L. H. and Hulek, K., A relative Hilbert–Mumford criterion, Manuscripta Math., to appear, doi:10.1007/s00229-015-0744-8.
[Hal10]Hall, J., Moduli of singular curves, Preprint (2010), arXiv:1011.6007.
[Har68]Harder, G., Halbeinfache gruppenschemata über vollständigen Kurven, Invent. Math. 6 (1968), 107149.
[Har77]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, 1977).
[Har10]Hartshorne, R., Deformation theory, Graduate Texts in Mathematics, vol. 257 (Springer, 2010).
[Has03]Hassett, B., Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), 316352.
[HS02]Hausel, T. and Sturmfels, B., Toric hyperkähler varieties, Doc. Math. 7 (2002), 495534.
[Hur11a]Huruguen, M., Compactification d’espaces homogènes sphériques sur un corps quelconque, PhD thesis, Université de Grenoble (2011), http://tel.archives-ouvertes.fr/docs/00/71/64/02/PDF/24054_HURUGUEN_2011_archivage.pdf.
[Hur11b]Huruguen, M., Toric varieties and spherical embeddings over an arbitrary field, J. Algebra 342 (2011), 212234.
[IM65]Iwahori, N. and Matsumoto, H., On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 548.
[dJo]de Jong, A. J. et al. , The stacks project, http://stacks.math.columbia.edu.
[Kau00]Kausz, I., A modular compactification of the general linear group, Doc. Math. 5 (2000), 553594.
[Kau05]Kausz, I., A Gieseker type degeneration of moduli stacks of vector bundles on curves, Trans. Amer. Math. Soc. 357 (2005), 48974955.
[KKMS73]Kempf, G., Knudsen, F., Mumford, D. and Saint-Donat, B., Toroidal embeddings I, Lecture Notes in Mathematics, vol. 339 (Springer, 1973).
[Kin94]King, A., Moduli of representations of finite-dimensional algebras, Q. J. Math. (2) 45 (1994), 515530.
[KT88]Kleiman, S. and Thorup, A., Complete bilinear forms, in Algebraic geometry (Sundance, UT, 1986), Lecture Notes in Mathematics, vol. 1311 (Springer, 1988).
[Knu71]Knutson, D., Algebraic spaces, Lecture Notes in Mathematics, vol. 203 (Springer, 1971).
[Kum02]Kumar, S., Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204 (Birkhäuser, 2002).
[Lak87]Laksov, D., Completed quadrics and linear maps, Algebraic geometry: Bowdoin 1985, Proceedings of Symposia in Pure Mathematics, vol. 46, Part 2, eds Bloch, Spencer et al. (American Mathematical Society, 1987), 321370.
[LM00]Laumon, G. and Moret-Bailly, L., Champs algébriques, Ergebnisse der Mathematik, vol. 39 (Springer, 2000).
[LM00]Losev, A. and Manin, Y., New moduli spaces of pointed curves and pencils of flat connections, Michigan Math. J. 48 (2000), 443472.
[Li01]Li, J., Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), 509578.
[MT12]Martens, J. and Thaddeus, M., Variations on a theme of Grothendieck, Preprint (2012), arXiv:1210.8161.
[Mil80]Milne, J., Étale cohomology (Princeton University Press, 1980).
[NS99]Nagaraj, D. S. and Seshadri, C. S., Degenerations of the moduli spaces of vector bundles on curves, II: Generalized Gieseker moduli spaces, Proc. Indian Acad. Sci. Math. Sci. 109 (1999), 165201.
[Pez10]Pezzini, G., Lectures on spherical and wonderful varieties, Les cours du CIRM 1 (2010), 3353, ccirm.cedram.org/ccirm-bin/fitem?id=CCIRM_2010__1_1_33_0.
[Ren05]Renner, L. E., Linear algebraic monoids, Encyclopaedia of Mathematical Sciences, vol. 134 (Springer, 2005).
[Rit98]Rittatore, A., Algebraic monoids and group embeddings, Transform. Groups 3 (1998), 375396.
[Rit01]Rittatore, A., Very flat reductive monoids, Publ. Mat. Urug. 9 (2001–2), 93121.
[Sch04]Schmitt, A., The Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves, J. Differential Geom. 66 (2004), 169209.
[Sem48]Semple, J. G., On complete quadrics, J. Lond. Math. Soc. 23 (1948), 258267.
[Sem51]Semple, J. G., The variety whose points represent complete collineations of S r on S r, Univ. Roma Ist. Naz. Alta Mat. Rend. Mat. e Appl. (5) 10 (1951), 201208.
[Sem52]Semple, J. G., On complete quadrics, II, J. Lond. Math. Soc. 27 (1952), 280287.
[Ser06]Sernesi, E., Deformations of algebraic schemes, Grundlehren Math. Wiss., vol. 334 (Springer, 2006).
[Spr06]Springer, T. A., Some results on compactifications of semisimple groups, Proceedings of the International Congress of Mathematicians, vol. II (European Mathematical Society, Zürich, 2006), 13371348.
[Ste65]Steinberg, R., Regular elements of semi-simple algebraic groups, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 4980.
[Teo02]Teodorescu, T., Principal bundles over chains or cycles of rational curves, Michigan Math. J. 50 (2002), 173186.
[Tha99]Thaddeus, M., Complete collineations revisited, Math. Ann. 315 (1999), 469495.
[Tim11]Timashev, D. A., Homogeneous spaces and equivariant embeddings, Encyclopaedia Math. Sci., vol. 138 (Springer, 2011).
[Vai84]Vainsencher, I., Complete collineations and blowing up determinantal ideals, Math. Ann. 267 (1984), 417432.
[Vin95]Vinberg, E. B., On reductive algebraic semigroups, in Lie groups and Lie algebras: E.B. Dynkin’s Seminar, American Mathematical Society Translations: Series 2, vol. 169 (American Mathematical Society, 1995), 145182.
[Vis05]Vistoli, A., Grothendieck topologies, fibered categories and descent theory, in Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs, vol. 123, eds Fantechi, B. et al. (American Mathematical Society, 2005).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Related content

Powered by UNSILO

Compactifications of reductive groups as moduli stacks of bundles

  • Johan Martens (a1) and Michael Thaddeus (a2)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.