Pluricomplex geometry and hyperbolic monopoles

Details Pluricomplex geometry and hyperbolic monopoles Pluricomplex geometry and hyperbolic monopoles

2011-04-12

arxiv.org/abs/1104.2270v2

Mathematics

Differential Geometry

33 pages; a superfluous assumption about purity of sheaves removed in section 2

Scientific paper

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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