Blowing up and desingularizing constant scalar curvature Kähler manifolds

By C. Arezzo, Frank Pacard. 2004

137 citations recorded by the source corpus.

This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow-ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature Kähler metrics.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ae6c8289ba642d2289c610986094ac56ffcf88ed

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Relative K-stability of extremal metrics (2009, methodology): “Building on the K-semistability proved by Donaldson [8] and on the work of Arezzo–Pacard [2] on blowing up cscK metrics, the first named author completed the proof of one direction of this conjecture, under the assumption that the automorphism group of (M,L) is discrete.”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, methodology): “We therefore avoid Arezzo-Pacard’s deep results on blow-ups of manifolds admitting cscK metrics [1].”
On the Calabi Flow (2006, methodology): “Around the same time, using method of gluing and grafting Arezzo-Pacard [2][3] were able to construct new CscK metrics on the blow up of manifold if the original manifold has a CscK metric.”
Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, methodology): “Using a blowup technique and the glueing theorem of Arezzo–Pacard [8], Stoppa [70] and Stoppa–Székelyhidi [72] have shown that the existence of a cscK or extremal Kähler metric in 2πc1(L) does actually imply (1,wex)-K-stability relative to T, i.”
Lectures on Stability and Constant Scalar Curvature (2008, background): “For example, there is a very rich literature on geometric constructions of metrics of constant scalar curvature, which we did not discuss, see e.g. [2, 3 , 75, 82, 116, 121].”
Filtrations and test-configurations (2011, methodology): “This will give us the required contradiction, since by Arezzo-Pacard’s result [2] the blowup admits a cscK metric for small ε, and so is asymptotically Chow stable by Donaldson’s result [8].”