Kaehler constant scalar curvature metrics on blow ups and resolutions of singularities

Abstract: After recalling the gluing construction for Kaehler constant scalar curvature and extremal (`a la Calabi) metrics starting from a compact or ALE orbifolds with isolated singularities, I will show how to compute the Futaki invariant of the adiabatic classes in this setting, extending previous work by Stoppa, Szekelyhidi and Odaka. Besides giving new existence and non-existence results, the connection with the Tian-Yau-Donaldson Conjecture and the K-stability of the resolved manifold will be discussed and the relevance towards the interpretation of the ADM mass for Kaehler manifolds.The original part of this talk will cover joint works with A. Della Vedova, R. Lena, K. Corrales and L. Mazzieri.