Maximal compatible splitting and diagonals of Kempf varieties

Lauritzen, Niels; Thomsen, Jesper Funch

Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2543-2575.

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.

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.

Classification : 14M15, 13A35
Keywords: Special linear group, Schubert variety, Frobenius splitting, maximal multiplicity, Wahl’s conjecture
Mot clés : groupe spécial linéaire, variétés de Schubert, scindage de Frobenius, multiplicité maximale, conjecture de Wahl

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