Maximal compatible splitting and diagonals of Kempf varieties

Abstract

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.

Degenerate coordinate rings of flag varieties and Frobenius splitting

“…It follows that σ splits the p th power morphism on R(L) and hence is a graded splitting of R(L). We also recall the following essential result from [1], [16].…”

“…By Lemma 3.1, H 0 ≥m (2(p − 1)ρ) is the subspace of H 0 (2(p − 1)ρ) consisting of elements that vanish to multiplicity at least m at the identity in G/B. It follows by Lemma 2.12 in[16] that a splitting section s ∈ H 0 (2(p − 1)ρ) maximally compatibly splits the identity if and only if s ∈ H 0 ≥(p−1)N (2(p − 1)ρ).…”

“…Let X be a nonsingular variety and let Y ⊆ X be a closed nonsingular variety with ideal sheaf I ⊆ O(X). Following Definition 2.2 in[16], we say that Y is maximally compatibly split if there is a Frobenius splitting σ : F * O X → O X such that σ I pm+1 ⊆ I m+1 for all m ≥ 0. (Remark that the equivalent condition proved in Lemma 2.12 of[16] is frequently given as the definition of a maximal compatible splitting, cf exercises 1.3.E.12 and 13 in[1].…”

A proof of Wahl's conjecture in the symplectic case

“…At the same time it was observed that this "canonical" Frobenius splitting would not work in general. Recently Lauritzen and Thomsen [6] introduced a new Frobenius splitting of X × X, for X = SLn(k) /B, with the desired vanishing property along the diagonal. As a consequence the LMP-conjecture is now verified for any generalized flag variety of the form SL n (k) /P .…”

Spherical varieties and Wahl’s conjecture

“…This conjecture has been considered by several authors (see for example [15], [12], [5], [13], [18]). In particular Brown and Lakshmibai in [5] proved this conjecture for minuscule homogeneous varieties using Representation Theoretic techniques and a case by case analysis.…”