Complex structures on nilpotent Lie algebras

We classify real 6-dimensional nilpotent Lie algebras for which the corresponding Lie group has a left-invariant complex structure, and estimate the dimensions of moduli spaces of such structures.Comment: Improved section 4, 20 pages, to appear in J Pure Appl Algebr

The intrinsic torsion of SU(3) and G_2 structures

We analyse the relationship between the components of the intrinsic torsion of an SU(3) structure on a 6-manifold and a G_2 structure on a 7-manifold. Various examples illustrate the type of SU(3) structure that can arise as a reduction of a metric with holonomy G_2.Comment: Proc. conf. Differential Geometry Valencia 200

Generalized Killing spinors in dimension 5

We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, SU(2)-structures on 5-manifolds defined by a generalized Killing spinor. We prove that in the real analytic case, such a 5-manifold can be isometrically embedded as a hypersurface in a Calabi-Yau manifold in a natural way. We classify nilmanifolds carrying invariant structures of this type, and present examples of the associated metrics with holonomy SU(3).Comment: 30 pages. v2: corrected the statement and proof of Theorem 14; added a comment on the embedding property in the non-real-analytic cas

Twistor Topology of the Fermat Cubic

We describe topologically the discriminant locus of a smooth cubic surface in the complex projective space ${\mathbb{CP}}^3$ that contains 5 fibres of the projection ${\mathbb{CP}}^3 \longrightarrow S^4$

Orthogonal complex structures on domains in R^4

An orthogonal complex structure on a domain in R^4 is a complex structure which is integrable and is compatible with the Euclidean metric. This gives rise to a first order system of partial differential equations which is conformally invariant. We prove two Liouville-type uniqueness theorems for solutions of this system, and use these to give an alternative proof of the classification of compact locally conformally flat Hermitian surfaces first proved by Pontecorvo. We also give a classification of non-degenerate quadrics in CP^3 under the action of the conformal group. Using this classification, we show that generic quadrics give rise to orthogonal complex structures defined on the complement of unknotted solid tori which are smoothly embedded in R^4.Comment: 42 pages. Version 2 contains several improvements and simplifications throughout. Material from the first version on more general branched coverings has been removed in order to make the article more focused, and will appear elsewher

Intersection numbers on moduli spaces and symmetries of a Verlinde formula

We investigate the geometry and topology of a standard moduli space of stable bundles on a Riemann surface, and use a generalization of the Verlinde formula to derive results on intersection pairings.Comment: AMSLaTex, 15 pages with 1 figur

Half-flat structures on S^3xS^3

We describe left-invariant half-flat SU(3)-structures on S^3xS^3 using the representation theory of SO(4) and matrix algebra. This leads to a systematic study of the associated cohomogeneity one Ricci-flat metrics with holonomy G_2 obtained on 7-manifolds with equidistant S^3xS^3 hypersurfaces. The generic case is analysed numerically.Comment: 23 pages, 6 figures. To appear in Annals of Global Analysis and Geometr

Bach-flat Lie groups in dimension 4

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.Comment: 4 page