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

Abstract

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.

Limits of Riemannian 4-manifolds and the symplectic geometry of their twistor spaces

“…In [18], Reznikov observed that Z also carries a natural closed 2-form ω, whose restriction to each fibre of π is the area form. Asking for ω to be symplectic gives a curvature inequality for g, which was first investigated by Reznikov [18] and later described explicitly in [11]. (In fact, this can be seen as a special case of the 'fat connections' introduced much earlier by Weinstein [19].)…”

Pure connection formulation, twistors, and the chase for a twistor action for general relativity

“…From this data only we show how to construct very natural structures on twistor space, namely the 1-form τ , some associated connection on O(n) bundle and the triplet (J A , ω A , g A ) of compatible almost complex structure, 2-form and Euclidean metric on twistor space of proposition 1. We also review, from [14], some symplectic structure that is naturally constructed from the connection. Finally we investigate the condition for integrability of the almost complex structure as well as the condition for which the triplet (J A , ω A , g A ) is Kähler.…”

“…This is a well known construction (cf [16]) and we here use the terminology of [14], [15]. We also briefly recall how to write equations for full Einstein gravity in terms of connections from [3], [25].…”

“…Symplectic structure on PT(M ) (Fine and Panov [14]) If A is a definite connection then ω s = (n − m) −1 d (n,m) 2 , n = m, is a symplectic structure on PT(M ).…”

“…Such connection can be assigned a sign. This terminology first appeared in [14] and we will review it in section 1. The possibility of associating a symplectic structure on PT(M ) with a definite SU (2)-connection on M was already pointed out in [14].…”