Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov [34]. We study this inequality in the case when the base has dimension four, with three main aims.Firstly, we use this approach to construct symplectic six-manifolds with c 1 = 0 which are never Kähler; e.g., we produce such manifolds with b 1 = 0 = b 3 and also with c 2 · [ω] < 0, answering questions posed by Smith-Thomas-Yau [37].Examples come from Riemannian geometry, via the Levi-Civita connection on Λ + . The underlying six-manifold is then the twistor space and often the symplectic structure tames the Eells-Salamon twistor almost complex structure. Our second aim is to exploit this to deduce new results about minimal surfaces: if a certain curvature inequality holds, it follows that the space of minimal surfaces (with fixed topological invariants) is compactifiable; the minimal surfaces must also satisfy an adjunction inequality, unifying and generalising results of Chen-Tian [6].One metric satisfying the curvature inequality is hyperbolic fourspace H 4 . Our final aim is to show that the corresponding symplectic manifold is symplectomorphic to the small resolution of the conifold xw − yz = 0 in C 4 . We explain how this fits into a hyperbolic description of the conifold transition, with isometries of H 4 acting symplectomorphically on the resolution and isometries of H 3 acting biholomorphically on the smoothing. * Supported by an FNRS postdoctoral fellowship. † Supported by EPSRC grant EP/E044859/1.

In this article we give a criterion for the existence of a metric of curvature 1 on a 2-sphere with n conical singularities of prescribed angles 2πϑ 1 , . . . , 2πϑ n and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in ϑ 1 , . . . , ϑ n .

We use hyperbolic geometry to construct simply connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kähler structure. We start with the desingularisations of the quadric cone in C 4 : the smoothing is a natural S 3 -bundle over H 3 , its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S 2 -bundle over H 4 with symplectic geometry determined by the metric. Using hyperbolic geometry, we find orbifold quotients with trivial canonical bundle; smooth examples are produced via crepant resolutions.In particular, we find the first example of a simply connected symplectic 6-manifold with c 1 D 0 that does not admit a compatible Kähler structure. We also find infinitely many distinct complex structures on 2.S 3 S 3 / # .S 2 S 4 / with trivial canonical bundle. Finally, we explain how an analogous construction for hyperbolic manifolds in higher dimensions gives symplectic non-Kähler "Fano" manifolds of dimension 12 and higher.53D35, 32Q55; 51M10, 57M25

In this article we introduce the notion of polyhedral Kähler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces and classify the singularities of the metric. The singularities form a divisor and the residues of the flat connection on the complement of the divisor give us a system of cohomological equations. A parabolic version of the Kobayshi-Hitchin correspondence of T Mochizuki permits us to characterize polyhedral Kähler metrics of nonnegative curvature on CP 2 with singularities at complex line arrangements.53C56; 32Q15, 53C55

No abstract