Jumps, folds and hypercomplex structures
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© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Roger Bielawski · Carolin Peternell
Jumps, folds and hypercomplex structures Received: 14 June 2019 / Accepted: 14 October 2019 Abstract. We investigate the geometry of the Kodaira moduli space M of sections of π : Z → P1 , the normal bundle of which is allowed to jump from O(1)n to O(1)n−2m ⊕ O(2)m ⊕ Om . In particular, we identify the natural assumptions which guarantee that the Obata connection of the hypercomplex part of M extends to a logarithmic connection on M.
1. Introduction It is well known that a hyperkähler or a hypercomplex structure on a smooth manifold M can be encoded in the twistor space, which is a complex manifold Z fibring over P1 and equipped with an antiholomorphic involution σ covering the antipodal map. The manifold M is recovered as the parameter space of σ -invariant sections with normal bundle isomorphic to O(1)⊕n (n = dimC M). If we start with an arbitrary complex manifold Z equipped with a holomorphic submersion π : Z → P1 and an involution σ , then the corresponding component of the Kodaira moduli spaceof sections of π will typically also contain sections with other normal bunn O(ki ). This kind of jumping normal bundle attracted recently attention dles i=1 in the case of 4-dimensional hyperkähler manifolds [3,4,6], in the context of a phenomenon known as folding (one speaks then of folded hyperkähler metrics). Folded hyperkähler structures do not exhaust all geometric possibilities which arise when the normal bundle is allowed to jump. Even in four dimensions there are examples which are not folded (Example 2.2 below). The aim of this paper is to investigate the natural extension of the hypercomplex geometry arising on such manifolds of sections (folded or not). More precisely, we are interested in the differential geometry of the (smooth) parameter space M of sections of π : Z → P1 n with normal bundle N isomorphic i=1 O(ki ), where each ki ≥ 0. We shall discuss only the purely holomorphic case, i.e. we are interested in all sections, not just σ -invariant. Choosing an appropriate σ allows one to carry over all results to hypercomplex or split hypercomplex manifolds. Both authors are members of, and the second author is fully supported by the DFG Priority Programme 2026 “Geometry at infinity”. R. Bielawski (B) · C. Peternell: Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany. e-mail: [email protected] Mathematics Subject Classification: 53C26 · 53C28
https://doi.org/10.1007/s00229-019-01160-7
R. Bielawski, C. Peternell
Our particular object of interest is the (holomorphic) Obata connection ∇, i.e. the unique torsion free connection preserving the hypercomplex (or, rather, the biquaternionic, i.e. complexified hypercomplex) structure. This is defined on the open subset U of M corresponding to the sections with normal bundle isomorphic to O(1)⊕n . The general twistor machinery (see, e.g. [1]) implies that ∇ extends to a first order differential ope
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