Pluricomplex geometry and hyperbolic monopoles

TL;DR

This paper introduces pluricomplex geometry, a generalization of hypercomplex geometry, to better understand the structure of moduli spaces of hyperbolic monopoles, revealing new geometric and twistor-theoretic properties.

Contribution

It defines pluricomplex geometry, explores its properties, and connects it to the geometry of hyperbolic monopoles and twistor theory.

Findings

Existence of a canonical torsion-free connection on pluricomplex manifolds

Pluricomplex structures parameterize algebraic curves of higher genera

Generalization of hypercomplex geometry with a 2-sphere of complex structures

Abstract

Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a 2-sphere of complex structures, but they no longer behave like unit imaginary quaternions. We still require, however, that the 2-sphere of complex structures determines a decomposition of the complexified tangent space as tensor product of C^{2n} and C^2. Among interesting properties of pluricomplex manifold is the existence of a canonical torsion free connection. Pluricomplex manifolds have also remarkable twistor theory: they parameterise algebraic curves of higher genera.