Okounkov bodies associated to pseudoeffective divisors

Abstract

An Okounkov body is a convex subset in Euclidean space associated to a big divisor on a smooth projective variety with respect to an admissible flag. In this paper, we introduce two convex bodies associated to pseudoeffective divisors, called the valuative Okounkov bodies and the limiting Okounkov bodies, and show that these convex bodies reflect the asymptotic properties of pseudoeffective divisors as in the case with big divisors. Our results extend the works of Lazarsfeld-Mustat ¸ȃ and Kaveh-Khovanskii. For this purpose, we define and study special subvarieties, called the Nakayama subvarieties and the positive volume subvarieties, associated to pseudoeffective divisors.

Restricted volumes on Kähler manifolds

“…which also shows that Vol X|V (L) is not in general a numerical invariant of the line bundle L (another example can be found in [12,Example 2.3]).…”

“…Restricted volumes have proved to be an extremely useful tool in algebraic geometry, see e.g. [5,8,10,12,19,20,28,32,29,36,38] and references therein.…”

“…By homogeneity, this result also holds when L and H are just Q-divisors. The quantity on the right hand side of (1.3) has been recently considered explicitly in [12]. In fact, more is true.…”

“…This is because c 1 (L) dim V X|V = 0 by definition, while Vol X|V (L) = 0 since B − (L) ⊂ B(L) (the stable base locus of L) and it is clear from the definition that Vol X|V (L) = 0 holds whenever V ⊂ B(L) (this observation also justifies our definition of α dim V X|V = 0 when V ⊂ E nn (α)). But when V ⊂ B + (L) in general we have that c 1 (L) dim V X|V = Vol X|V (L), see Example 2.12 (or [12,Example 2.3] for another example). Note that this example shows that Vol X|V (L) is not in general a numerical invariant of the line bundle L, while obviously…”

“…Remark 2.11. The quantity on the RHS of (1.3) has also recently been studied in [12], where it is denoted by Vol + X|V (L) (see their Definition 2.2). Example 2.12.…”