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Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold

par Fine, Joel ;Panov, Dmitri
Référence Journal of Differential Geometry, 82, 1, page (155-205)
Publication Publié, 2009

Article révisé par les pairs

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Résumé :

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 Rezn- ikov [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 [w] < 0, answering questions posed by Smith-Thomas-Yau [37]. Examples come from Riemannian geometry, via the Levi-Civita connection on A +. 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 four-space 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.