Fixed divisors on hyperkähler manifolds

TL;DR

This work analyzes base divisors of nef and big divisors on hyperkähler manifolds by examining the fixed and movable parts of linear systems through the Beauville–Bogomolov form. Building on vanishing theorems, the strictly increasing Riemann–Roch polynomial, and Markman-type divisibility, the authors prove that the fixed part of |A| is reduced and that |2A| is movable; in dimension four they obtain a stronger decomposition A ≅ dL + B with B negative in BB-form and, after birational modification, a basepoint-free |L| that induces a Lagrangian fibration onto a projective plane. The results connect fixed divisors to Lagrangian fibrations without relying on the full hyperkähler SYZ conjecture and provide a detailed numerical description of the fixed and movable parts via the BB-form. This advances understanding of the geometry of nef/big divisors on hyperkähler manifolds and highlights when a fixed divisor forces a fibration structure in low dimension.

Abstract

Let $X$ be a hyperkähler manifold, and let $A$ be a nef and big divisor on $X$. We show that the fixed part of the linear system $|A|$ is reduced and as a consequence $|2A|$ is mobile. If $X$ has dimension four we also show that if the fixed part of $|A|$ is not empty, the mobile part induces a (rational) Lagrangian fibration.

Theorems & Definitions (13)

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