Positivity in the Grothendieck group of complex flag varieties

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.