  We prove a conjecture of A. S. Buch concerning the structure constants of the
Grothendieck ring of a flag variety with respect to its basis of Schubert
structure sheaves. For this, we show that the coefficients in this basis of the
structure sheaf of any subvariety with rational singularities, have alternating
signs. Equivalently, the class of the dualizing sheaf of such a subvariety is a
nonnegative combination of classes of dualizing sheaves of Schubert varieties.
