Smooth structures and Einstein metrics on

Abstract

ABSTRACT. We show that each of the topological 4-manifolds CP 2 #kCP 2 , for k = 6, 7 admits a smooth structure which has an Einstein metric of scalar curvature s > 0, a smooth structure which has an Einstein metric with s < 0 and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We show that there are infinitely many manifolds homeomorphic non-diffeomorphic to CP 2 #5CP 2 which don't admit an Einstein metric. We also exhibit new examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature. The main ingredients are recent constructions of exotic symplectic or complex manifolds with small topological numbers.

The Yamabe invariant of a class of symplectic 4-manifolds

“…In general, it is difficult to determine if the associated manifold N is Kähler or not. In particular cases, as the examples in [19,25], one can show that a global Q−Gorenstein smoothing of M exists and conclude that the manifold N is in fact Kähler. In Sect.…”

“…Assume that the canonical sheaf K M ′ is ample. Then, the symplectic manifold N obtained by taking the generalized rational blowdown of the chains C i in M is a minimal, symplectic 4−manifold of general type, and its Yamabe invariant Y (N ) is [19,25]), or an elliptic surface, as the family of examples in Sect. 4.…”

A simply connected surface of general type withpg= 0 andK2= 4

“…We explain it in a detail in the rest of this section. Proof of Theorem 4.1 Based on the idea Rȃsdeaconu and Ş uvaina [9], we show that the surface X t has an ample canonical bundle. Then it follows from Theorem 4.2 of Aubin-Yau that there exists a Kähler-Einstein metric on X t of negative scalar curvature.…”

“…Recently Rȃsdeaconu and Ş uvaina [9] proved the existence of a smooth structure on each of the topological 4-manifolds CP 2 ] kCP 2 , for k D 6; 7, which has an Einstein metric of negative scalar curvature. By applying their method on the surface X t constructed in Section 2, we can easily prove the existence of a Kähler-Einstein metric on X t with negative scalar curvature.…”

“…The main result of this paper is the following theorem. Remark Rȃsdeaconu and Ş uvaina [9] proved that the complex surfaces constructed in Lee and Park [5] and Park-Park-Shin [6] admit Kähler-Einstein metrics of negative scalar curvature. By applying their method to the complex surface constructed in this paper, one may prove that it also admits a Kähler-Einstein metric of negative scalar curvature; see Section 4.…”