Abelian fibred holomorphic symplectic manifolds

TL;DR

This paper explores holomorphic symplectic manifolds fibred by abelian varieties, aiming to understand their structure and contribute to classification efforts through geometric analysis and conjectures.

Contribution

It introduces the study of abelian fibred holomorphic symplectic manifolds as a higher-dimensional analogue of elliptic fibrations on K3 surfaces, proposing new conjectures and geometric insights.

Findings

Identification of conditions for fibering in abelian varieties

Formulation of conjectures on the structure of these fibrations

Potential pathways toward classification of holomorphic symplectic manifolds

Abstract

We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in this way, and are led to several natural conjectures. We then study the geometry of these fibrations. The expectation is that this point of view will prove useful in understanding holomorphic symplectic manifolds, and possibly lead to a classification.