 
Summary: OKOUNKOV BODIES AND TORIC DEGENERATIONS
DAVE ANDERSON
Abstract. Let be the Okounkov body of a divisor D on a projec
tive variety X. We describe a geometric criterion for to be a lattice
polytope, and show that in this situation X admits a flat degeneration
to the corresponding toric variety. This degeneration is functorial in an
appropriate sense.
As an application, we construct a toric degeneration of the (general
ized) flag variety G/B for each reduced word for w, compatible with
a toric degeneration of the corresponding BottSamelson variety, using
very little representation theory as input.
1. Introduction
Let X be a projective algebraic variety of dimension d over an algebraically
closed field k, and let D be a big divisor on X. (Following [LM], all divisors
are Cartier in this article.) As part of his proof of the logconcavity of
the multiplicity function for representations of a reductive group, Okounkov
showed how to associate to D a convex body (D) Rd [Ok1, Ok2]. The
construction depends on a choice of flag Y· of subvarieties of X, that is, a
chain X = Y0 Y1 · · · Yd, where Yi is a subvariety of codimension i in
X which is nonsingular at the point Yd. Very roughly, one uses Y· to define
