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Differential geometry of Hilbert schemes of curves in a projective space. (English) Zbl 1432.53068

The authors introduce the notion of a quaternionic \(r\)-Kronecker structure of rank \(k\) on a manifold \(M\). This is a certain type of bundle map \(\alpha:E\otimes \mathbb{C}^r\to T^{\mathbb{C}}M\), where \(E\) is a quaternionic vector bundle of rank \(k\). In the case where \(\alpha\) is an isomorphism and \(r=2\), this reduces to an almost hypercomplex structure. They also introduce a complex analog, as well as twistor spaces of (integrable and regular) Kronecker structures. These are complex manifolds admitting a natural holomorphic submersion to \(\mathbb{C}\mathrm{P}^{r-1}\) from which the original manifold may be recovered as a suitable space of sections, analogous to the so-called twistor lines in the hypercomplex case. Conversely, it is proven that for any complex manifold \(Z\) with a surjective holomorphic submersion \(\pi:Z\to \mathbb{C}\mathrm{P}^{r-1}\), certain spaces of sections carry natural \(r\)-Kronecker structures.
After discussing connections to hypercomplex geometry, the authors apply these concepts to certain open subsets \(M_{d,g}\) of the Hilbert scheme of curves of fixed genus \(g\) and degree \(d\) in \(\mathbb{C}\mathrm{P}^3\), showing that the subspace \(M_{d,g}^\sigma\) of real curves admits a quaternionic \(4\)-Kronecker structure of rank \(2d\) (though it may be empty for some values of \(d,g\)). They also consider curves in \(\mathbb{C}\mathrm{P}^n\) for \(n>3\), where the assumptions needed to ensure smoothness impose stronger constraints on \(d\) and \(g\). In the final section, the authors return to the case \(n=3\), and study \(M_{1,0}^\sigma=S^4\) in more detail.

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C28 Twistor methods in differential geometry
14C05 Parametrization (Chow and Hilbert schemes)
14H50 Plane and space curves
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References:

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