Okounkov bodies and toric degenerations

Abstract

Let Δbe the Okounkov body of a divisor D on a projective variety X. We describe a geometric criterion for Δto 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.

v3: final version, some expository improvements in Section 5, to appear in Math. Ann