Section 1 (1.1) Let V be a three-dimensional vector space over an algebraically closed field K. Let X be the projective plane of lines in V and let Y be the dual projective plane of planes in V. The flag variety F is the subvariety of X x Y consisting of pairs (l, s) where the line is contained in the plane s. F is a homogeneous space under the action of SL(V). Let n x (respectively rr) be the projection X x Y X (respectively X x Y Y). Let [(i, j) ryx(gx(i (R) r(gr(j), for any pair of integers

more »

... i, j) which is a line bundle on X x Y. Let La(i, j)denote its restriction to F. A line bundle is called singular if any of the following conditions hold: i=-1, j 1, or + j -2. If a line bundle is non-singula, r, its index is defined to be the number of negative integers in the set {i + 1, j + 1, + j + 2}. There is a general theorem of Bott [5] giving the structure of the cohomology of flag varieties if char (K) 0. In the case of F it is: THEOREM 1.1. Let char (K)= 0. The cohomology n(oL'(i, j)):# (0) iff is non-singular and q is the index of '. vector space The purpose of this paper is to determine the analogous theorem when char (K)= p > 0. The following theorem is a special case of a theorem of Kempf [12]. THEOREM 1.2. Let char (K) be arbitrary. Assume q is either 0 or 3. The following are equivalent: (i) Ur('(i, j))=/: (0) if and only if r q. (ii) .'(i, j) is non-singular of index q. The next theorem is the major result of this paper. It shows that Bott's theorem is false in positive characteristic. THEOREM 1.3. a<p. Assume char (K)= p. Let a and b be positive integers with