Complex Symplectic Lie Algebras with Large Abelian Subalgebras

TL;DR

This paper classifies complex symplectic structures on certain Lie algebras and constructs examples of complex symplectic manifolds, including those lacking hyperk"ahler metrics and featuring Lagrangian torus fibrations.

Contribution

It provides a complete classification of complex symplectic structures on almost abelian Lie algebras and constructs new examples of complex symplectic manifolds with specific geometric properties.

Findings

Classified complex symplectic structures on almost abelian Lie algebras.

Constructed examples of complex symplectic manifolds without hyperk"ahler metrics.

Produced manifolds with Lagrangian torus fibrations.

Abstract

We present two constructions of complex symplectic structures on Lie algebras with large abelian ideals. In particular, we completely classify complex symplectic structures on almost abelian Lie algebras. By considering compact quotients of their corresponding connected, simply connected Lie groups we obtain many examples of complex symplectic manifolds which do not carry (hyper)k\"ahler metrics. We also produce examples of compact complex symplectic manifolds endowed with a fibration whose fibers are Lagrangian tori.