Maximal compatible splitting and diagonals of Kempf varieties
Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 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: 10.5802/aif.2682
Classification: 14M15, 13A35
Keywords: Special linear group, Schubert variety, Frobenius splitting, maximal multiplicity, Wahl’s conjecture
Lauritzen, Niels 1; Thomsen, Jesper Funch 1

1 Aarhus University Department of Mathematical Sciences Bygning 1530 Ny Munkegade 118 8000 Århus C (Denmark)
     author = {Lauritzen, Niels and Thomsen, Jesper Funch},
     title = {Maximal compatible splitting and diagonals of {Kempf} varieties},
     journal = {Annales de l'Institut Fourier},
     pages = {2543--2575},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {6},
     year = {2011},
     doi = {10.5802/aif.2682},
     mrnumber = {2976320},
     zbl = {1251.14037},
     language = {en},
     url = {}
AU  - Lauritzen, Niels
AU  - Thomsen, Jesper Funch
TI  - Maximal compatible splitting and diagonals of Kempf varieties
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 2543
EP  - 2575
VL  - 61
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  -
UR  -
UR  -
UR  -
DO  - 10.5802/aif.2682
LA  - en
ID  - AIF_2011__61_6_2543_0
ER  - 
%0 Journal Article
%A Lauritzen, Niels
%A Thomsen, Jesper Funch
%T Maximal compatible splitting and diagonals of Kempf varieties
%J Annales de l'Institut Fourier
%D 2011
%P 2543-2575
%V 61
%N 6
%I Association des Annales de l’institut Fourier
%R 10.5802/aif.2682
%G en
%F 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

Cited by Sources: