Zariski F-decomposition and Lagrangian fibration on hyperk\"ahler manifolds

TL;DR

This paper studies Zariski decompositions on hyperk"ahler manifolds, provides conditions for Lagrangian fibrations, and proves the termination of D-flops sequences, advancing understanding of their birational geometry.

Contribution

It introduces a Zariski decomposition for pseudo-effective divisors and establishes criteria for hyperk"ahler manifolds to admit Lagrangian fibrations, also proving finite termination of D-flops.

Findings

Zariski decomposition exists for pseudo-effective divisors on hyperk"ahler manifolds

Provides a sufficient condition for Lagrangian fibrations

D-flops sequences terminate after finitely many steps

Abstract

For a compact hyperk\"ahler manifold X, we show certain Zariski decomposition for every pseudo-effective R-divisor, and give a sufficient condition for X to be bimeromorphic to a (holomorphic) Lagrangian fibration. We also prove that any sequence of D-flops between projective hyperk\"ahler manifolds terminates after finite steps.