The generalized Mukai conjecture for symmetric varieties

Gagliardi, Giuliano; Hofscheier, Johannes

doi:10.1090/tran/6738

The generalized Mukai conjecture for symmetric varieties
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by Giuliano Gagliardi and Johannes Hofscheier PDF

Trans. Amer. Math. Soc. 369 (2017), 2615-2649

Abstract:

We associate to any complete spherical variety $X$ a certain nonnegative rational number $\wp ({X})$, which we conjecture to satisfy the inequality $\wp ({X}) \le \dim X - \mathrm {rank} X$ with equality holding if and only if $X$ is isomorphic to a toric variety. We show that, for spherical varieties, our conjecture implies the generalized Mukai conjecture on the pseudo-index of smooth Fano varieties due to Bonavero, Casagrande, Debarre, and Druel. We also deduce from our conjecture a smoothness criterion for spherical varieties. It follows from the work of Pasquier that our conjecture holds for horospherical varieties. We are able to prove our conjecture for symmetric varieties.

References

Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
Victor V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), no. 3, 493–535. MR 1269718
Laurent Bonavero, Cinzia Casagrande, Olivier Debarre, and Stéphane Druel, Sur une conjecture de Mukai, Comment. Math. Helv. 78 (2003), no. 3, 601–626 (French, with English and French summaries). MR 1998396, DOI 10.1007/s00014-003-0765-x
P. Bravi and D. Luna, An introduction to wonderful varieties with many examples of type $\rm F_4$, J. Algebra 329 (2011), 4–51. MR 2769314, DOI 10.1016/j.jalgebra.2010.01.025
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
P. Bravi and G. Pezzini, Wonderful subgroups of reductive groups and spherical systems, J. Algebra 409 (2014), 101–147. MR 3198836, DOI 10.1016/j.jalgebra.2014.03.018
P. Bravi and G. Pezzini, The spherical systems of the wonderful reductive subgroups, J. Lie Theory 25 (2015), no. 1, 105–123. MR 3345829
Chal Benson and Gail Ratcliff, A classification of multiplicity free actions, J. Algebra 181 (1996), no. 1, 152–186. MR 1382030, DOI 10.1006/jabr.1996.0113
P. Bravi, Primitive spherical systems, Trans. Amer. Math. Soc. 365 (2013), no. 1, 361–407. MR 2984062, DOI 10.1090/S0002-9947-2012-05621-2
Michel Brion, Vers une généralisation des espaces symétriques, J. Algebra 134 (1990), no. 1, 115–143 (French). MR 1068418, DOI 10.1016/0021-8693(90)90214-9
Michel Brion, Variétés sphériques et théorie de Mori, Duke Math. J. 72 (1993), no. 2, 369–404 (French). MR 1248677, DOI 10.1215/S0012-7094-93-07213-4
M. Brion, Curves and divisors in spherical varieties, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 21–34. MR 1635672
Michel Brion, The total coordinate ring of a wonderful variety, J. Algebra 313 (2007), no. 1, 61–99. MR 2326138, DOI 10.1016/j.jalgebra.2006.12.022
Paolo Bravi and Bart Van Steirteghem, The moduli scheme of affine spherical varieties with a free weight monoid, Int. Math. Res. Not. IMRN, electronically published on October 5, 2015, DOI: http://dx.doi.org/10.1093/imrn/rnv281 (to appear in print).
Romain Camus, Variétés sphériques affines lisses, Thèse de doctorat, Université Joseph Fourier, 2001.
Cinzia Casagrande, The number of vertices of a Fano polytope, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 1, 121–130 (English, with English and French summaries). MR 2228683
Stéphanie Cupit-Foutou, Wonderful varieties: a geometrical realization, arXiv:0907.2852v4.
David A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17–50. MR 1299003
C. De Concini and C. Procesi, Complete symmetric varieties, Invariant theory (Montecatini, 1982) Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 1–44. MR 718125, DOI 10.1007/BFb0063234
Giuliano Gagliardi, The Cox ring of a spherical embedding, J. Algebra 397 (2014), 548–569. MR 3119238, DOI 10.1016/j.jalgebra.2013.08.037
Giuliano Gagliardi, A combinatorial smoothness criterion for spherical varieties, Manuscripta Math. 146 (2015), no. 3-4, 445–461. MR 3312454, DOI 10.1007/s00229-014-0713-7
Giuliano Gagliardi and Johannes Hofscheier, Gorenstein spherical Fano varieties, Geom. Dedicata 178 (2015), 111–133. MR 3397485, DOI 10.1007/s10711-015-0047-y
Giuliano Gagliardi and Johannes Hofscheier, Homogeneous spherical data of orbits in spherical embeddings, Transform. Groups 20 (2015), no. 1, 83–98. MR 3317796, DOI 10.1007/s00031-014-9297-2
Friedrich Knop, The Luna-Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) Manoj Prakashan, Madras, 1991, pp. 225–249. MR 1131314
Hanspeter Kraft and Vladimir L. Popov, Semisimple group actions on the three-dimensional affine space are linear, Comment. Math. Helv. 60 (1985), no. 3, 466–479. MR 814152, DOI 10.1007/BF02567428
Andrew S. Leahy, A classification of multiplicity free representations, J. Lie Theory 8 (1998), no. 2, 367–391. MR 1650378
Ivan V. Losev, Uniqueness property for spherical homogeneous spaces, Duke Math. J. 147 (2009), no. 2, 315–343. MR 2495078, DOI 10.1215/00127094-2009-013
D. Luna, Grosses cellules pour les variétés sphériques, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 267–280 (French). MR 1635686
D. Luna, Variétés sphériques de type $A$, Publ. Math. Inst. Hautes Études Sci. 94 (2001), 161–226 (French). MR 1896179, DOI 10.1007/s10240-001-8194-0
D. Luna and Th. Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), no. 2, 186–245 (French). MR 705534, DOI 10.1007/BF02564633
Boris Pasquier, Variétés horosphériques de Fano, Bull. Soc. Math. France 136 (2008), no. 2, 195–225 (French, with English and French summaries). MR 2415341, DOI 10.24033/bsmf.2554
Boris Pasquier, The pseudo-index of horospherical Fano varieties, Internat. J. Math. 21 (2010), no. 9, 1147–1156. MR 2725271, DOI 10.1142/S0129167X10006422
Nicolas Perrin, On the geometry of spherical varieties, Transform. Groups 19 (2014), no. 1, 171–223. MR 3177371, DOI 10.1007/s00031-014-9254-0
Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
Dmitry A. Timashev, Homogeneous spaces and equivariant embeddings, Encyclopaedia of Mathematical Sciences, vol. 138, Springer, Heidelberg, 2011. Invariant Theory and Algebraic Transformation Groups, 8. MR 2797018, DOI 10.1007/978-3-642-18399-7
Thierry Vust, Plongements d’espaces symétriques algébriques: une classification, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 2, 165–195 (French). MR 1076251