The algebra of exterior differential forms on a regular 3-Sasakian 7-manifold is investigated, with special reference to nearly-parallel G_2 3-forms. This is applied to the study of 3-forms invariant under cohomogeneity-one actions by SO(4) on the 7-sphere and on Berger's space SO(5)/SO(3).

A study of tensors on the quaternionic projective plane HP2 arising from a stable 3-form on C6 and an associated action of SU(3) is related to the existence of a holomorphic rank 3 vector bundle over CP5 discovered by Horrocks. It also leads to the construction of SU(3) invariant Spin(7) structures on HP2, which are characterised in terms of associated 4-forms.

A study is made of algebraic curves and surfaces in the flag manifold 𝔽=SU(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 .

A study is made of left-invariant G(2)-structures with an exact 3-form on a Lie group G whose Lie algebra g admits a codimension-one nilpotent ideal h. It is shown that such a Lie group G cannot admit a left-invariant closed G(2)-eigenform for the Laplacian and that any compact solvmanifold Gamma/G arising from G does not admit an (invariant) exact G(2)-structure. We also classify the seven-dimensional Lie algebras g with codimension-one ideal equal to the complex Heisenberg Lie algebra which admit exact G(2)-structures with or without special torsion. To achieve these goals, we first determine the six-dimensional nilpotent Lie algebras h admitting an exact SL(3, C)-structure rho or a half-flat SU (3)-structure (omega, rho) with exact rho, respectively.

A study is made of left-invariant $\mathrm{G}_2$-structures with an exact 3-form on a Lie group $G$ whose Lie algebra $\mathfrak{g}$ admits a codimension-one nilpotent ideal $\mathfrak{h}$. It is shown that such a Lie group $G$ cannot admit a left-invariant closed $\mathrm{G}_2$-eigenform for the Laplacian and that any compact solvmanifold $Γ\backslash G$ arising from $G$ does not admit an (invariant) exact $\mathrm{G}_2$-structure. We also classify the seven-dimensional Lie algebras $\mathfrak{g}$ with codimension-one ideal equal to the complex Heisenberg Lie algebra which admit exact $\mathrm{G}_2$-structures with or without special torsion. To achieve these goals, we first determine the six-dimensional nilpotent Lie algebras $\mathfrak{h}$ admitting an exact $\mathrm{SL}(3,\mathbb{C})$-structure $ρ$ or a half-flat $\mathrm{SU}(3)$-structure $(ω,ρ)$ with exact $ρ$, respectively.

SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on $\mathbb C^d$. The resulting equations are interpreted in terms of moment maps by focussing attention on the orbit of a cyclic subgroup and the maximal torus in $\mathrm U(d)$ that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.

A study is made of R6 as a singular quotient of the conical space R+×CP3 with holonomy G2, with respect to an obvious action by U(1) on CP3 with fixed points. Closed expressions are found for the induced metric, and for both the curvature and symplectic 2-forms characterizing the reduction. All these tensors are invariant by a diagonal action of SO(3) on R6, which can be used effectively to describe the resulting geometrical features.

We describe the 8-dimensional Wolf spaces as cohomogeneity one SU(3)-manifolds, and discover perturbations of the quaternionkähler metric on the simply-connected 8-manifold G2/SO(4) that carry a closed fundamental 4-form but are not Einstein. To Nigel Hitchin on the occasion of his 70th birthday

We describe the 8-dimensional Wolf spaces as cohomogeneity one SU(3)-manifolds, and discover perturbations of the quaternion-kaehler metric on the simply-connected 8-manifold G_2/SO(4) that carry a closed fundamental 4-form but are not Einstein.

This is a survey of results concerning special and exceptional Riemannian holonomy from a historical and personal perspective.

We classify SIC-POVMs of rank one in CP^2, or equivalently sets of nine equally-spaced points in CP^2, without the assumption of group covariance. If two points are fixed, the remaining seven must lie on a pinched torus that a standard moment mapping projects to a circle in R^3. We use this approach to prove that any SIC set in CP^2 is isometric to a known solution, given by nine points lying in triples on the equators of the three 2-spheres each defined by the vanishing of one homogeneous coordinate. We set up a system of equations to describe hexagons in CP^2 with the property that any two vertices are related by a cross ratio (transition probability) of 1/4. We then symmetrize the equations, factor out by the known solutions, and compute a Groebner basis to show that no SIC sets remain. We do find new configurations of nine points in which 27 of the 36 pairs of vertices of the configuration are equally spaced.

We develop a calculus of differential forms on a quaternion-Kähler manifold M4n admitting an isometric circle action. This is used to study three fundamental examples of such actions on the quaternionic projective plane and the construction of G2 and half-flat structures on quotients of M8 and its hypersurfaces.

We describe topologically the discriminant locus of a smooth cubic surface in the complex projective space CP 3 that contains 5 fibres of the projection CP 3 ! S 4 .

The theory of slice-regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains \Omega of \mathbb R^4 . When \Omega is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which \Omega is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space \mathbb CP^3 .

We describe left-invariant half-flat -structures on using the representation theory of and matrix algebra. This leads to a systematic study of the associated cohomogeneity one Ricci-flat metrics with holonomy obtained on -manifolds with equidistant hypersurfaces. The generic case is analysed numerically.

We describe left-invariant half-flat \( \mathrm{SU }(3) \)-structures on \( S^3\times S^3\) using the representation theory of \( \mathrm SO (4) \) and matrix algebra. This leads to a systematic study of the associated cohomogeneity one Ricci-flat metrics with holonomy \( \mathrm G _2\) obtained on \( 7 \)-manifolds with equidistant \( S^3\times S^3\) hypersurfaces. The generic case is analysed numerically.

This paper pursues the study of the Calabi-Yau equation on certain symplectic non-Kaehler 4-manifolds, building on a key example of Tosatti-Weinkove in which more general theory had proved less effective. Symplectic 4-manifolds admitting a 2-torus fibration over a 2-torus base are modelled on one of three solvable Lie groups. Having assigned an invariant almost-Kaehler structure and a volume form that effectively varies only on the base, one seeks a symplectic form with this volume. Our approach simplifies the previous analysis of the problem, and establishes the existence of solutions in various other cases.

We establish the existence of solvable Lie groups of dimension 4 and left-invariant Riemannian metrics with zero Bach tensor which are neither conformally Einstein nor half conformally flat.