We construct a Kähler structure (which we call a generalised Kähler cone) on an open subset of the cone of a strongly pseudo-convex CR manifold endowed with a one-parameter family of compatible Sasaki structures. We determine those generalised Kähler cones which are Bochner-flat and we study their local geometry. We prove that any Bochner-flat Kähler manifold of complex dimension bigger than two is locally isomorphic to a generalised Kähler cone.

The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.

We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non-trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way.

We give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point m of a Dirac manifold M, there is a well-defined transverse Poisson structure to the pre-symplectic leaf P through m. Finally, we describe the neighborhood of a pre-symplectic leaf in terms of geometric data. This description agrees with that given by Vorobjev for the Poisson case.

Let Mμ0 denote S2×S2 endowed with a split symplectic form normalized so that μ≥1 and σ(S2)=1. Given a symplectic embedding of the standard ball of capacity c∈(0,1) into Mμ0, consider the corresponding symplectic blow-up . In this paper, we study the homotopy type of the symplectomorphism group and that of the space of unparametrized symplectic embeddings of Bc into Mμ0. Writing ℓ for the largest integer strictly smaller than μ, and λ∈(0,1] for the difference μ−ℓ, we show that the symplectomorphism group of a blow-up of ‘small’ capacity c<λ is homotopically equivalent to the stabilizer of a point in Symp(Mμ0), while that of a blow-up of ‘large’ capacity c≥λ is homotopically equivalent to the stabilizer of a point in the symplectomorphism group of a non-trivial bundle obtained by blowing down . It follows that, for c<λ, the space is homotopy equivalent to S2 ×S2, while, for c≥λ, it is not homotopy equivalent to any finite CW-complex. A similar result holds for symplectic ruled manifolds diffeomorphic to . By contrast, we show that the embedding spaces and , if non-empty, are always homotopy equivalent to the spaces of ordered configurations and . Our method relies on the theory of pseudo-holomorphic curves in 4 -manifolds, on the computation of Gromov invariants in rational 4 -manifolds, and on the inflation technique of Lalonde and McDuff.