Exotic Holonomy on Moduli Spaces of Rational Curves
Quo-Shin Chi, Lorenz J Schwachh\"ofer

TL;DR
This paper explores the existence of exotic holonomy connections on moduli spaces of rational curves, constructing new examples and analyzing their torsion properties, revealing novel holonomy representations.
Contribution
It constructs a torsion-free connection on the moduli space of rational contact curves and introduces a new exotic holonomy related to a six-dimensional representation of SL(2)×SL(2).
Findings
Existence of torsion-free connections with exotic holonomy on moduli spaces.
Construction of a connection on the space of all rational curves with specific torsion properties.
Identification of a new exotic holonomy as a six-dimensional representation of SL(2)×SL(2).
Abstract
Bryant \cite{Br} proved the existence of torsion free connections with exotic holonomy, i.e. with holonomy that does not occur on the classical list of Berger \cite{Ber}. These connections occur on moduli spaces of rational contact curves in a contact threefold . Therefore, they are naturally contained in the moduli space of all rational curves in . We construct a connection on whose restriction to is torsion free. However, the connection on has torsion unless both and are flat. We also show the existence of a new exotic holonomy which is a certain sixdimensional representation of . We show that every regular -connection (cf. \cite{Br}) is the restriction of a unique connection with this holonomy.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Algebraic Geometry and Number Theory
