Boucksom-Zariski and Weyl chambers on irreducible holomorphic symplectic manifolds

TL;DR

This paper studies the structure of cones of divisors on projective irreducible holomorphic symplectic manifolds, introducing chamber decompositions that clarify the behavior of the volume function and the pseudo-effective cone.

Contribution

It provides a detailed chamber decomposition of the big cone, linking Zariski and Weyl chambers, and describes the structure of the pseudo-effective cone for IHS manifolds.

Findings

Decomposition of the big cone into chambers with constant negative part support

Description of the volume function via chamber structure

Determination of the pseudo-effective cone structure

Abstract

Inspired by the work of Bauer, K\"uronya, and Szemberg, we provide for the big cone of a projective irreducible holomorphic symplectic (IHS) manifold a decomposition into chambers (which we describe in detail), in each of which the support of the negative part of the divisorial Zariski decomposition is constant. We see how the obtained decomposition of the big cone allows us to describe the volume function. Moreover, similarly to the case of surfaces, we see that the big cone of a projective IHS manifold admits a decomposition into simple Weyl chambers, which we compare to that induced by the divisorial Zariski decomposition. To conclude, we determine the structure of the pseudo-effective cone.