Polygons of Newton-Okounkov type on irreducible holomorphic symplectic manifolds

Abstract

Let $X$ be a projective irreducible holomorphic symplectic manifold. We associate with any big $\mathbf{R}$-divisor $D$ on $X$ a convex polygon ${\Delta }_E^{\mathrm{num}}(D)$ of dimension 2, whose Euclidean volume is ${\mathrm{vol}}_{{\mathbf{R}}^2}({\Delta }_E^{\mathrm{num}}(D))=q_X(P(D))/2$, where $E$ is any prime divisor on $X$, $q_X$ is the Beauville-Bogomolov-Fujiki form, and $P(D)$ is the positive part of the divisorial Zariski decomposition of $D$. We systematically study these polygons and observe that they behave like the Newton-Okounkov bodies of big divisors on smooth complex projective surfaces, with respect to a general admissible flag.

Final version. Accepted for publication in "Annali della Scuola Normale Superiore di Pisa - Classe di Scienze". 20 pages, minor corrections, and 2 figures were added