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Lars Sektnan : Blowing up extremal Poincaré type manifolds

One of the central conjectures in Kähler geometry is the Yau-Tian-Donaldson conjecture relating the existence of canonical Kähler metrics to algebro-geometric stability. A natural question is to ask what happens when such a metric does not exist, and here Kähler metrics of Poincaré type are expected to play an important role. These metrics are Kähler metrics defined on the complement of a divisor in a compact complex manifold and have a cusp-like singularity near the divisor. The blow-up theorem of Arezzo-Pacard and its generalizations give sufficient conditions for the blow-up of a compact Kähler manifold admitting a canonical metric to also carry such a metric. I will describe an extension of this result to the Poincaré type setting.

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