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Let X = X ¯ − D be a smooth quasi-projective curve. We previously constructed a Deligne–Hitchin moduli space with Hecke gauge groupoid for connections of rank 2. We extend this construction to the case of any rank r , although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure. [ABSTRACT FROM AUTHOR]

In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that C P 3 is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold. [ABSTRACT FROM AUTHOR]

A study is made of algebraic curves and surfaces in the flag manifold F = S U (3) / T 2 , and their configuration relative to the twistor projection π from F to the complex projective plane P 2 , defined with the help of an anti-holomorphic involution j . This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space P 3 of the 4-dimensional sphere S 4 . Deformations of twistor fibers project to real surfaces in P 2 , whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in F of bidegree (1 , 1) . These surfaces define orthogonal complex structures on specified dense open subsets of P 2 relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree (1 , 1) are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in F . [ABSTRACT FROM AUTHOR]

Let X be a quasi-projective curve, compactified to (Y, D) with X = Y - D . We construct a Deligne–Hitchin twistor space out of moduli spaces of framed λ -connections of rank 2 over Y with logarithmic singularities and quasi-parabolic structure along D. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on X give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights 0, 1, 2. The weight 2 piece corresponds to the deformations of the KMS structure including parabolic weights and the residues of the λ -connection. [ABSTRACT FROM AUTHOR]

A hyperkähler manifold M has a family of induced complex structures indexed by a two-dimensional sphere S 2 ≅ CP 1 . The twistor space of M is a complex manifold Tw (M) together with a natural holomorphic projection Tw (M) → CP 1 , whose fiber over each point of CP 1 is a copy of M with the corresponding induced complex structure. We remove one point from this sphere (corresponding to one fiber in the twistor space), and for the case of M = R 4 n , n ∈ N , equipped with the standard hyperkähler structure, we construct one quantization that replaces the family of Berezin–Toeplitz quantizations parametrized by S 2 - { p t } . We provide semiclassical asymptotics for this quantization. [ABSTRACT FROM AUTHOR]

We study the holomorphic vector bundles E over the twistor space Tw (M) of a compact simply connected hyperkähler manifold M. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection π : Tw (M) ⟶ CP 1 . It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on Tw (M) of prime rank, we prove that its restriction to the generic fibre of π is stable. On the other hand, for a K3 surface M, we construct examples of irreducible vector bundles of any composite rank on Tw (M) whose restriction to every fibre of π is non-stable. We have obtained a new method of constructing irreducible vector bundles on hyperkähler twistor spaces; this method is employed in constructing these examples. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the construction of Deligne–Hitchin twistor space by gluing two certain Hodge moduli spaces. We investigate some properties of such generalized Deligne–Hitchin twistor space as a complex analytic manifold. More precisely, we show it admits holomorphic sections whose normal bundle contains a semistable subbundle with positive degree and whose energy is semi-negative, and it carries a balanced metric. Moreover, we also study the automorphism groups of the Hodge moduli spaces and the generalized Deligne–Hitchin twistor spaces. [ABSTRACT FROM AUTHOR]

A study of tensors on the quaternionic projective plane HP 2 arising from a stable 3-form on C 6 and an associated action of SU (3) is related to the existence of a holomorphic rank 3 vector bundle over CP 5 discovered by Horrocks. It also leads to the construction of SU (3) invariant Spin (7) structures on HP 2 , which are characterised in terms of associated 4-forms. [ABSTRACT FROM AUTHOR]