Okounkov bodies and toric degenerations
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Mathematische Annalen
Okounkov bodies and toric degenerations Dave Anderson
Received: 21 June 2012 / Revised: 5 October 2012 / Published online: 30 November 2012 © Springer-Verlag Berlin Heidelberg 2012
Abstract Let Δ be the Okounkov body of a divisor D on a projective variety X . We describe a geometric criterion for Δ 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. 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 [22], all divisors are Cartier in this article.) As part of his proof of the log-concavity of the multiplicity function for representations of a reductive group, Okounkov showed how to associate to D a convex body ΔY• (D) ⊆ Rd [23,24]. The construction depends on a choice of flag Y• of subvarieties of X , that is, a chain X = Y0 ⊃ Y1 ⊃ · · · ⊃ Yd , where Yr is a subvariety of codimension r in X which is nonsingular at the point Yd . One uses the flag to define a valuation ν = νY• , which in turn defines a graded semigroup ΓY• ⊆ N × Nd ; the convex body Δ = ΔY• (D) is the intersection of {1} × Rd with the closure of the convex hull of ΓY• in R × Rd . The details of this construction will be reviewed in Sect. 3.
The author was partially supported by NSF Grants DMS-0502170 and DMS-0902967, and also by the Clay Mathematics Institute as a Liftoff Fellow. D. Anderson (B) Department of Mathematics, University of Washington, Seattle, WA 98195, USA e-mail: [email protected]; [email protected] Present Addresss: D. Anderson Institut de Mathématiques de Jussieu, 75013 Paris, France
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Recently, Kaveh–Khovanskii [18] and Lazarsfeld–Musta¸ta˘ [22] have systematically developed this construction, and exploited it to show that ΔY• (D) captures much of the geometry of D. For example, the volume of D (as a divisor) is equal to the Euclidean volume of Δ (up to a normalizing factor of d!), and one can use this to prove continuity of the volume function, as a map N 1 (X )R → R (see [22, Theorem B] and the references given there). Many intersection-theoretic notions can also be defined and generalized using the convex bodies Δ(D); this is discussed at length in [18]. These Okounkov bodies—as ΔY• (D) is called in the literature stemming from [22]—are generally quite difficult to compute. They are often not polyhedral; when polyhedral, they are often not rational; and even if Δ is a rational polyhedron, the semigroup used to define it need not be finitely generated. In fact, the numbers appearing as volumes or coordinates of Okounkov bodies can be quite general [20]; they certainly may be irrational. One situation, however, is easy. When X is a smooth toric variety, D is a T -invariant ample divisor, and Y• is a flag of T -invariant subvarieties, the Okounkov body ΔY• (D) is the lattice polytope associated to D by the usual correspondence of toric geome
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