Pluricomplex geometry and hyperbolic monopoles

Roger Bielawski; Lorenz Schwachhöfer

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arXiv · 1104.2270

Pluricomplex geometry and hyperbolic monopoles

Roger Bielawski·Lorenz Schwachhöfer

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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.

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Roger Bielawski, Lorenz Schwachhöfer. 2011-04-12. Pluricomplex geometry and hyperbolic monopoles. https://arxiv.org/abs/1104.2270

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