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.

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Abstract. We provide infinitely many examples of pairs of diffeomorphic, non-simply-connected Kähler manifolds of complex dimension three with different Kodaira dimensions. Also, in any possible Kodaira dimension we find infinitely many pairs of non-deformation equivalent, diffeomorphic Kähler threefolds.

Abstract.We find an obstruction to the existence of non-singular solutions to the normalized Ricci flow on four-manifolds with b + = 1. By using this obstruction, we study the relationship between the existence or non-existence of non-singular solutions of the normalized Ricci flow and exotic smooth structures on the topological 4-manifolds CP 2 # CP 2 , where 5 ≤ ≤ 8. Mathematics Subject Classification (2000). Primary 53C21; Secondary 57R55.

We explore the maximality of the Hilbert square of maximal real surfaces, and find that in many cases the Hilbert square is maximal if and only if the surface has connected real locus. In particular, the Hilbert square of no maximal K3-surface is maximal. Nevertheless, we exhibit maximal surfaces with disconnected real locus whose Hilbert square is maximal.On dédaigne volontiers un but qu'on n'a pas réussi à atteindre, ou qu'on a atteint définitivement.M. Proust, A la recherche du temps perdu.