The Yamabe invariant of a class of symplectic 4-manifolds

Abstract

We compute the Yamabe invariant for a class of symplectic 4−manifolds of general type obtained by taking the rational blowdown of Kähler surfaces. In particular, for any point on the half-Noether line we exhibit a simply connected minimal symplectic manifold for which we compute the Yamabe invariant.

On the Scalar Curvature of 4-Manifolds

“…However, this turned out to be wrong. Indeed, Ioana S ¸uvaina [28] later succeeded in proving that certain symplectic manifolds arising from Gompf's rational-blow-down construction have Yamabe invariants that do exactly saturate (43), even though they do not admit Kähler metrics. In her examples, the rational-blow-down operation can be carried out by starting with a Kähler-Einstein orbifold, and then gluing in finite quotients of Gibbons-Hawking gravitational instantons [27].…”