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Polygons of Newton-Okounkov type on irreducible holomorphic symplectic manifolds

Francesco Antonio Denisi

TL;DR

The paper develops a Newton-Okounkov-type polygonal framework for projective irreducible holomorphic symplectic (IHS) manifolds by associating, to a big $\mathbf{R}$-divisor $D$ and a prime divisor $E$, a convex polygon $\Delta_E^{\mathrm{num}}(D)$ whose area equals $q_X(P(D))/2$, with $P(D)$ the positive part of the divisorial Zariski decomposition. It proves convexity and piecewise-linearity of these polygons via Boucksom-Zariski chambers, shows a two-dimensional fiber structure over $\mathrm{Big}(X)$, and establishes Minkowski decompositions with respect to a finite Minkowski basis under rational polyhedrality assumptions. The constructions connect restricted volumes to the BBF form and provide a systematic, computable approach to capture positivity data on IHS manifolds beyond the classical higher-dimensional Newton-Okounkov bodies. The paper also furnishes explicit examples, including Hilbert schemes of K3s and a cubic fourfold Fano variety, illustrating both rational and irrational polygonal behavior, and highlighting the scope and limits of the Minkowski framework. This contributes a practical, 2D convex-geometry toolkit for positivity on IHS manifolds with potential computational applications and connections to birational geometry.

Abstract

Let $X$ be a projective irreducible holomorphic symplectic manifold. We associate with any big $\mathbf{R}$-divisor $D$ on $X$ a convex polygon $Δ_E^{\mathrm{num}}(D)$ of dimension 2, whose Euclidean volume is $\mathrm{vol}_{\mathbf{R}^2}(Δ_E^{\mathrm{num}}(D))=q_X(P(D))/2$, 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.

Polygons of Newton-Okounkov type on irreducible holomorphic symplectic manifolds

TL;DR

The paper develops a Newton-Okounkov-type polygonal framework for projective irreducible holomorphic symplectic (IHS) manifolds by associating, to a big -divisor and a prime divisor , a convex polygon whose area equals , with the positive part of the divisorial Zariski decomposition. It proves convexity and piecewise-linearity of these polygons via Boucksom-Zariski chambers, shows a two-dimensional fiber structure over , and establishes Minkowski decompositions with respect to a finite Minkowski basis under rational polyhedrality assumptions. The constructions connect restricted volumes to the BBF form and provide a systematic, computable approach to capture positivity data on IHS manifolds beyond the classical higher-dimensional Newton-Okounkov bodies. The paper also furnishes explicit examples, including Hilbert schemes of K3s and a cubic fourfold Fano variety, illustrating both rational and irrational polygonal behavior, and highlighting the scope and limits of the Minkowski framework. This contributes a practical, 2D convex-geometry toolkit for positivity on IHS manifolds with potential computational applications and connections to birational geometry.

Abstract

Let be a projective irreducible holomorphic symplectic manifold. We associate with any big -divisor on a convex polygon of dimension 2, whose Euclidean volume is , where is any prime divisor on , is the Beauville-Bogomolov-Fujiki form, and is the positive part of the divisorial Zariski decomposition of . 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.
Paper Structure (11 sections, 18 theorems, 72 equations, 2 figures)

This paper contains 11 sections, 18 theorems, 72 equations, 2 figures.

Table of Contents

  1. Conventions
  2. Introduction
  3. Preliminaries
  4. Asymptotic base loci
  5. Volumes
  6. Divisorial Zariski decompositions
  7. Restricting volumes along prime divisors
  8. Polygons of Newton-Okounkov-type
  9. Minkowski decompositions
  10. Constructing a Minkowski basis
  11. Examples

Key Result

Theorem A

The subset $\Delta_E^{\mathrm{num}}(D)$ of $\mathbf{R}^2$ introduced above is a convex polygon. Moreover, $\mathrm{vol}_{\mathbf{R}^2}(\Delta_E^{\mathrm{num}}(D))=q_X(P(D))/2$, where $\mathrm{vol}_{\mathbf{R}^2}(\Delta_E^{\mathrm{num}}(D))$ is the Euclidean volume of the polygon $\Delta_E^{\mathrm{n where $\Delta(D)$ is the classical Newton Okounkov body of $D$, with respect to any admissible flag

Figures (2)

  • Figure 1: The polygon $\Delta_{E'}^{\mathrm{num}}(3H-E)$.
  • Figure 2: The polygon $\Delta_{\tau}^{\mathrm{num}}(g)$.

Theorems & Definitions (56)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 46 more