105 research outputs found
Asymptotic Chow polystability in K\"ahler geometry
It is conjectured that the existence of constant scalar curvature K\"ahler
metrics will be equivalent to K-stability, or K-polystability depending on
terminology (Yau-Tian-Donaldson conjecture).
There is another GIT stability condition, called the asymptotic Chow
polystability.
This condition implies the existence of balanced metrics for polarized
manifolds for all large .
It is expected that the balanced metrics converge to a constant scalar
curvature metric as tends to infinity under further suitable stability
conditions. In this survey article I will report on recent results saying that
the asymptotic Chow polystability does not hold for certain constant scalar
curvature K\"ahler manifolds. We also compare a paper of Ono with that of Della
Vedova and Zuddas.Comment: Survey paper submitted to the Proceedings of ICCM 201
The weighted Laplacians on real and complex metric measure spaces
In this short note we compare the weighted Laplacians on real and complex
(K\"ahler) metric measure spaces. In the compact case K\"ahler metric measure
spaces are considered on Fano manifolds for the study of K\"ahler-Einstein
metrics while real metric measure spaces are considered with Bakry-\'Emery
Ricci tensor. There are twisted Laplacians which are useful in both cases but
look alike each other. We see that if we consider noncompact complete manifolds
significant differences appear.Comment: Minor modifications. Submitted to Shoshichi Kobayashi memorial volum
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