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A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphisms and closed 2-forms. In the special case of six dimensions we characterize them as critical points of a natural variational problem on closed forms, and prove that a local moduli space is provided by an open set in either the odd or even cohomology.

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We describe two constructions of hyperkahler manifolds, one based on a Legendre transform, and one on a symplectic quotient. These constructions arose in the context of supersymmetric nonlinear σ-models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.

The Hitchin component of the character variety of representations of a surface group π 1 (S) into PSL d (R) for some d ≥ 3 can be equipped with a pressure metric whose restriction to the Fuchsian locus equals the Weil-Petersson metric up to a constant factor. We show that if the genus of S is at least 3, then the Fuchsian locus contains quasi-convex subsets of infinite diameter for the Weil-Petersson metric whose diameter for the path metric of the pressure metric is finite. This is established through showing that biinfinite paths of bending deformations have controlled bounded length. To this end we give a geometric interpretation of Fock-Goncharov positivity and show that bending deformations of Fuchsian representations stabilize a uniform Finsler quasi-convex disk in the symmetric space PSL d (R)/PSO(d).