Basic remarks on Lagrangian submanifolds of Hyper-K\"ahler manifolds

TL;DR

This paper discusses fundamental topological restrictions on Lagrangian surfaces within hyper-K"ahler 4-folds and explores their interaction with Lagrangian fibrations, contributing to the understanding of hyper-K"ahler geometry.

Contribution

It provides new basic restrictions on the topology of Lagrangian surfaces and insights into their interaction with Lagrangian fibrations in hyper-K"ahler manifolds.

Findings

Topological restrictions on Lagrangian surfaces in hyper-K"ahler 4-folds

Remarks on the interaction with Lagrangian fibrations

Insights into the structure of hyper-K"ahler manifolds

Abstract

This note presents basic restrictions on the topology "general" Lagrangian surfaces of hyper-K\"ahler $4$-folds and a remark on the interaction of a Lagrangian subvariety with a Lagrangian fibration of the associated hyper-K\"ahler variety.