4.1. Theorem ("[15], Theorem 3). — Let the base field be of arbitrary characteristic. Let L, M be line bundles on G/Q and X C G/Q a Schubert variety. Then
(i) Any Schubert variety is normal.
(ii) 7/'H°(G/Q,L)+0^rfH°(G/Q,M)4=0 then H°(X, L) ®H°(X, M) ^H°(X,L®M) is surjective.
(iii) If L is ample on G/Q then in the projective embedding given by L, X is projectively normal.
Proof. — We sketch the proof from [15]. We can assume X == X^CG/B. Find a simple root a such that ^(ojj = ^((0) — 1- Then X^ C X^ and TT : G/B -> G/P^
maps X^, birationally onto 7r(XJ over which X^ is a P^bundle. By induction X^, is normal and since TT, fi^ === ^(x ) (Theorem 3.8), Tr(X^) is normal and hence so is X^.
(ii) and (iii) The proof of these in [15] uses some results on Steinberg modules. The methods of this paper on the Frobenius splitting of the diagonal (Theorem 3.5) give an alternative proof. See Theorem 3.11 (ii) and Corollary 2.3. (One goes to charac-teristic p by semicontinuity.)
In characteristic zero any proper birational trivial map ^ '" Z -> X with Z smooth automatically satisfies R1 ^ K^ == 0, i > 0 by a theorem of Grauert-Riemenschneider.
Hence X is then Cohen-Macaulay by a result ofKempf ([16], Proposition 4). Therefore since any Schubert variety X C G/B admits a trivial resolution (Theorem 3.9) this gives a proof for the Gohen-Macaulayness of any Schubert variety in characteristic zero.
This remark should be attributed to Kempf, see Demazure [3], § 5, Corollary 2. (Thus the first proof of the Cohen-Macaulayness for Schubert varieties in characteristic zero comes from the result of [12] and [18], see Remark 4.5 below.) For arbitrary charac-teristic we have the following result.
4.2. Theorem (Cf. [16], Theorem 5). — Let the base field be of arbitrary characteristic.
(i) Any Schubert variety is Cohen-Macaulay.
(ii) In any Rrojective embedding given by an ample line bundle on G/Q any Schubert variety is arithmetically Cohen-Macaulay.
(iii) The canonical sheaf of a Schubert variety X C G/B is Igx® ^x("~~ D n X) where Igx is the ideal sheaf in X of ^X = union of all the codimension 1 Schubert subvarieties of X and D n X is the divisor cut out on X by the divisor D in G/B (see § 3.4).
Proof. — For the proof of (i) and (ii) see [16]. The assertion (iii) follows from the proof of Theorem 4 [16] by looking at the kernel of ^f* 0^— SZ,) -> ^ <r, ^(— ffL!) in the notation of that paper.
We take this opportunity to point out that for the proof the claim A, ^. i in the course of the proof of Theorem 4 in [16] we use by induction not only A^ but also B,.
Unfortunately this is not made clear there, causing some obscurity.
The result (iii) will probably help one towards a combinatorial criterion for a Schubert variety to be Gorenstein.
4.3. Remark. — Instead of working with the split exact sequence
0 - ^ o / B ^ F ^ G / B - ^ C - ^ O
one could tensor it with suitable line bundles and taking global sections one could work with the resulting sequences of G-modules (or B-modules). This way it is possible to work out proofs for all the results stated in this paper in a language which is less geo-metric and more suited to representation theory.
4.4. Remarks. — Using the inductive machinary of standard resolutions it is not difficult to see the following implications. These remarks are essentially due to Seshadri.
See [17], [18], [19]. Let X C G/B be a Schubert variety over a given field k.
(i) Normality of Schubert varieties is equivalent to the validity of the character formula of Demazure for large powers of an ample line bundle.
(ii) The result H^X, L) == 0, i > 0 and H°(G/B, L) -> H°(X, L) surjective for ample L and the normality of Schubert varieties together imply the same results for effective L and Demazure's character formula.
(iii) H^X, L) = 0 , i > 0, H°(G/B, L) -> H°(X, L) surjective for effective L => Nor-mality of Schubert varieties and Demazure's character formula.
Joseph proved Demazure's character formula when char k •== 0. (Demazure's proof in [3] contains an error (Proposition 11) as pointed out by V. Kac.) Therefore by (i) this implies the normality of Schubert varieties in characteristic 0.
Seshadri ([18], [19]) proved the normality of Schubert varieties in arbitrary characteristic. Hence by (ii) the results of [12] together with this give Demazure's character formula for arbitrary base fields and prove his conjecture. This is the first proof of Demazure's conjecture and justification of his work over arbitrary fields.
In [15] the result H^X, L) = 0, H°(G/B, L) -> H°(X, L) surjective for effective L is proved. By (iii) this again gives a simple and complete justification of Demazure's work and his conjecture over arbitrary fields.
In addition the Frobenius method gives the projective normality [15] and arith-metic Gohen-Macaulay property [16] of Schubert varieties and the results of the present paper.
Andersen in his preprint [I], which is later than [18] but earlier than [15], claimed a proof of the normality of Schubert varieties and Demazure's character formula. But in his proof he assumed without proof (see page 9, line 3 of [1]), that H^X^L) =H^(7^:(X),L) where TC:G/Q,->G/Q; and X a Schubert variety (Theorem 3.8 above). As is evident this is far from being an obvious fact; in fact one can quickly deduce normality etc. from this.
But this problem with [1] does not affect his paper [2] because the key point of [2], namely the splitting map 6 of the lemma of § 2 in [2], is not in [1]: Andersen was influenced by the paper [15] of Ramanan and Ramanathan.
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School of Mathematics
Tata Institute of Fundamental Research Bombay 400005.
Department of Mathematics The Johns Hopkins University Baltimore MD 21218.
Manuscrit refu Ie 14 aoUt 1985.