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{\it Name}: BRION Michel
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{\it Title}: Positivity in the Grothendieck group of flag varieties.
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{\it Abstract}: The Hilbert polynomial of a projective variety $Y$ is
a linear combination of Hilbert polynomials of projective subspaces,
with integer coefficients. If $Y$ is nonsingular and the ground field
has characteristic zero, then the signs of these coefficients are
alternating (as follows easily from the Kodaira vanishing theorem).
We generalize this result to subvarieties $Y$ of any flag variety $X$,
by replacing the Hilbert polynomial of $Y$ with the class of the
structure sheaf ${\cal O}_Y$ in the Grothendieck ring $K(X)$. As an
application, we solve a conjecture of A. S. Buch (math.AG/0004137)
concerning the signs of the structure constants of $K(X)$ with respect
to its basis of Schubert structure sheaves.
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