Weighted cscK metrics on Kähler varieties

Abstract

We study the weighted constant scalar curvature Kähler equations on mildly singular Kähler varieties. Assuming the existence of a suitable resolution of singularities, we establish the existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive for an extremal weight. This extends the works of Chen-Cheng and He to the singular weighted setting. Moreover, we provide a method for constructing examples of singular cscK metrics inspired by the work of Arezzo-Pacard. In contrast to the usual gluing techniques, our approach does not require a precise understanding about of the metric behavior near the singular locus.

arXiv comment35 pages; v3: corrected inaccuracies in the earlier version, accepted by J. Lond. Math. Soc

References in corpus (11)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• Extremal metrics on blow ups
• On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties
• Existence and deformations of Kahler-Einstein metrics on smoothable Q-Fano varieties
• K-stability of cubic threefolds
• K-semistability of cscK manifolds with transcendental cohomology class
• Kähler metrics with constant weighted scalar curvature and weighted K-stability
• Equidistribution for sequences of line bundles on normal Kaehler spaces
• Geometry and Topology of the space of Kähler metrics on singular varieties
• Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics
• Kähler-Einstein metrics on families of Fano varieties