Given an SO(3)-bundle with connection, the associated twosphere
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 c1 = 0 which are never K¨ahler; e.g., we produce such
manifolds with b1 = 0 = b3 and also with c2 · [ω] < 0, answering
questions posed by Smith–Thomas–Yau [37].
Examples come from Riemannian geometry, via the Levi–Civita
connection on +. 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 H4. Our final aim is to show that the corresponding
symplectic manifold is symplectomorphic to the small resolution
of the conifold xw −yz = 0 in C4. We explain how this fits into a
hyperbolic description of the conifold transition, with isometries of
H4 acting symplectomorphically on the resolution and isometries
of H3 acting biholomorphically on the smoothing.