Hyperkähler manifolds of curves in twistor spaces (Q2447891)

The author is concerned with constructions of hypercomplex and hyper-Kähler structures from curves of arbitrary degree. These constructions are motivated by the new construction of \textit{O. Nash} [Nonlinearity 20, No. 7, 1645--1675 (2007; Zbl 1130.81049)] who gave a new twistor construction of hyper-Kähler metrics on moduli spaces of \(\mathrm{SU}(2)\) magnetic monopoles, the so-called generalized Legendre transform of hyper-Kähler metrics due to \textit{U. Lindström} and \textit{M. Roček} [Commun. Math. Phys. 115, No. 1, 21--29 (1988; Zbl 0639.53077)] and the well-known fact that the smooth locus of the Hilbert scheme of curves of degree \(d\) and genus \(g\) in \(\mathbb{P}^3\) has, if nonempty, dimension \(4d\). There is a common framework for the three situations presented above. Let \(M\) be a connected hypercomplex or a hyper-Kähler manifold. The twistor space of \(M\) is a complex manifolds \(Z\) fibring over \(\mathbb{P}^1\), and equipped with an antiholomorphic involution \(\sigma\) which covers the antipodal map on \(\mathbb{P}^1\). If one considers \(\sigma\)-invariant curves of higher degree in \(Z\), one obtains a hypercomplex manifold, as long as one requires that the normal bundle of such a curve satisfies a certain ``stability condition''. The newly obtained hypercomplex manifold is (pseudo)-hyper-Kähler if \(M\) was so. The differential geometry of hyper-Kähler manifolds obtained from higher-degree curves is richer than just hyper-Kähler geometry. The author presents some descriptions of natural objects on the obtained manifold directly in terms of the complex manifold \(Z\). He constructs the canonical connection via a canonical splitting on the level sections of vector bundles, without a corresponding splitting of vector bundles on \(Z\).