This article finds constant scalar curvature Kähler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to curve, with fibres of genus at least 2. The proof is via an adiabatic limit. An approximate solution is constructed out of the hyperbolic metrics on the fibres and a large multiple of a certain metric on the base. A parameter dependent inverse function theorem is then used to perturb the approximate solution to a genuine solution in the same cohomology class. The arguments also apply to certain higher dimensional fibred Kähler manifolds.
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.
Let X ⊂ CP N be a smooth subvariety. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X which attempts to deform the given embedding into a balanced one. If L → X is an ample line bundle, considering embeddings via H 0 (L k ) gives a sequence of balancing flows. We prove that, provided these flows are started at appropriate points, they converge to Calabi flow for as long as it exists. This result is the parabolic analogue of Donaldson's theorem relating balanced embeddings to metrics with constant scalar curvature [13]. In our proof we combine Donaldson's techniques with an asymptotic result of Liu-Ma [17] which, as we explain, describes the asymptotic behaviour of the derivative of the map FS • Hilb whose fixed points are balanced metrics. * Supported by an FNRS postdoctoral fellowship.
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
A hypersymplectic structure on a 4-manifold X is a triple ω of symplectic forms which at every point span a maximal positive-definite subspace of Λ 2 for the wedge product. This article is motivated by a conjecture of Donaldson: when X is compact ω can be deformed through cohomologous hypersymplectic structures to a hyperkähler triple. We approach this via a link with G2-geometry. A hypersymplectic structure ω on a compact manifold X defines a natural G2-structure φ on X × T 3 which has vanishing torsion precisely when ω is a hyperkähler triple. We study the G2-Laplacian flow starting from φ, which we interpret as a flow of hypersymplectic structures. Our main result is that the flow extends as long as the scalar curvature of the corresponding G2-structure remains bounded. An application of our result is a lower bound for the maximal existence time of the flow, in terms of weak bounds on the initial data (and with no assumption that scalar curvature is bounded along the flow). Preliminary definitions and short time existenceThe main result of this section is:Proposition 2.1. Let ω be a hypersymplectic structure on a compact 4-manifold X. Then there exists a unique short time solution ω(t) to the hypersymplectic flow (3) starting at ω.This will follow easily from the analogous result of Bryant-Xu for the G 2 -Laplacian flow. Along the way we make explicit the relationship between the natural metric g ω on X induced by a Proof. This is immediate from the uniqueness part of Bryant-Xu's theorem [4, Theorem 0.1]: since the initial data φ(0) is T 3 invariant, so is the ensuing flow.By T 3 -invariance, the 3-form φ(t) on X × T 3 necessarily has the shape∈ Ω 2 (X) and D(t) ∈ Ω 3 (X) are paths of forms on X. Moreover, since dφ(t) = 0, it follows that these forms on X are closed.Lemma 2.7. A(t) = 1.Proof. We know that dA(t) = 0, i.e., that for every t, A(t) is constant. Moreover,Since ∂ t φ is exact, this integral is independent of t, and so A(t) = A(0) = 1.Next we consider the involution ϑ : X × T 3 → X × T 3 given by ϑ(p, t) → (p, −t) and writê φ(t) = −ϑ * φ(t).Lemma 2.8. For all t,φ(t) = φ(t). Hence B i (t) = 0 = D(t) vanish identically, and φ(t) remains of the form (2) for as long as it exists, for a closed triple ω(t) of 2-forms on X.
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