Maximal compatible splitting and diagonals of Kempf varieties
Annales de l'Institut Fourier, Volume 61 (2011) no. 6, p. 2543-2575

Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles, of blow-ups along the diagonal in flag varieties along with the LMP and Wahl conjectures in positive characteristic for the special linear group.

Lakshmibai, Mehta et Parameswaran (LMP) ont introduit la notion de multiplicité maximale dans le scindage de Frobenius.

Dans cet article, nous définissons l’analogue algébrique de cette notion et nous construisons un scindage de Frobenius avec multiplicité maximale le long de la diagonale de la variété des drapeaux complets.

Notre scindage induit aussi un scindage diagonal avec multiplicité maximale pour une classe particulière de variétés de Schubert lisses introduite par Kempf.

Comme conséquences, nous obtenons des scindages de Frobenius des fibrés tangents et des éclatements le long des diagonales dans les variétés de drapeaux, ainsi que les conjectures de LMP et de Wahl en caractéristique positive pour le groupe spécial linéaire.

DOI : https://doi.org/10.5802/aif.2682
Classification:  14M15,  13A35
Keywords: Special linear group, Schubert variety, Frobenius splitting, maximal multiplicity, Wahl’s conjecture
@article{AIF_2011__61_6_2543_0,
     author = {Lauritzen, Niels and Thomsen, Jesper Funch},
     title = {Maximal compatible splitting and diagonals of Kempf varieties},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {6},
     year = {2011},
     pages = {2543-2575},
     doi = {10.5802/aif.2682},
     mrnumber = {2976320},
     zbl = {1251.14037},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_6_2543_0}
}
Lauritzen, Niels; Thomsen, Jesper Funch. Maximal compatible splitting and diagonals of Kempf varieties. Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 2543-2575. doi : 10.5802/aif.2682. http://www.numdam.org/item/AIF_2011__61_6_2543_0/

[1] Brion, Michel Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, Birkhäuser, Basel (Trends Math.) (2005), pp. 33-85 | MR 2143072

[2] Brion, Michel; Kumar, Shrawan Frobenius splitting methods in geometry and representation theory, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 231 (2005) | MR 2107324 | Zbl 1072.14066

[3] Brown, J.; Lakshmibai, V. Wahl’s conjecture for a minuscule G/P, Proc. Indian Acad. Sci. Math. Sci., Tome 119 (2009) no. 5, pp. 571-592 | Article | MR 2598420 | Zbl 1192.14036

[4] Harris, Joe Algebraic geometry, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 133 (1992) (A first course) | MR 1182558 | Zbl 0779.14001

[5] Hartshorne, Robin Ample subvarieties of algebraic varieties, Springer-Verlag, Berlin, Notes written in collaboration with C. Musili. Lecture Notes in Mathematics, Vol. 156 (1970) | MR 282977 | Zbl 0208.48901

[6] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York (1977) (Graduate Texts in Mathematics, No. 52) | MR 463157 | Zbl 0531.14001

[7] Kempf, George R. Vanishing theorems for flag manifolds, Amer. J. Math., Tome 98 (1976) no. 2, pp. 325-331 | Article | MR 409493 | Zbl 0338.14019

[8] Kumar, Shrawan Proof of Wahl’s conjecture on surjectivity of the Gaussian map for flag varieties, Amer. J. Math., Tome 114 (1992) no. 6, pp. 1201-1220 | Article | MR 1198300 | Zbl 0790.14015

[9] Kumar, Shrawan; Lauritzen, Niels; Thomsen, Jesper Funch Frobenius splitting of cotangent bundles of flag varieties, Invent. Math., Tome 136 (1999) no. 3, pp. 603-621 | Article | MR 1695207 | Zbl 0959.14031

[10] Lakshmibai, V. Kempf varieties, J. Indian Math. Soc. (N.S.), Tome 40 (1976) no. 1-4, p. 299-349 (1977) | MR 506317 | Zbl 0447.14014

[11] Lakshmibai, V.; Mehta, V. B.; Parameswaran, A. J. Frobenius splittings and blow-ups, J. Algebra, Tome 208 (1998) no. 1, pp. 101-128 | Article | MR 1643983 | Zbl 0955.14006

[12] Lakshmibai, Venkatramani; Raghavan, Komaranapuram N.; Sankaran, Parameswaran Wahl’s conjecture holds in odd characteristics for symplectic and orthogonal Grassmannians, Cent. Eur. J. Math., Tome 7 (2009) no. 2, pp. 214-223 | Article | MR 2506962 | Zbl 1200.14100

[13] Mehta, V. B.; Parameswaran, A. J. On Wahl’s conjecture for the Grassmannians in positive characteristic, Internat. J. Math., Tome 8 (1997) no. 4, pp. 495-498 | Article | MR 1460897 | Zbl 0914.14021

[14] Mehta, V. B.; Ramanathan, A. Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2), Tome 122 (1985) no. 1, pp. 27-40 | Article | MR 799251 | Zbl 0601.14043

[15] Mumford, David The red book of varieties and schemes, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1358 (1999) (Includes the Michigan lectures (1974) on curves and their Jacobians, With contributions by Enrico Arbarello) | MR 1748380 | Zbl 0658.14001

[16] Wahl, Jonathan Gaussian maps and tensor products of irreducible representations, Manuscripta Math., Tome 73 (1991) no. 3, pp. 229-259 | Article | MR 1132139 | Zbl 0764.20022

[17] Zariski, Oscar; Samuel, Pierre Commutative algebra. Vol. II, Springer-Verlag, New York (1975) (Reprint of the 1960 edition, Graduate Texts in Mathematics, Vol. 29) | MR 389876 | Zbl 0313.13001