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 $(M, L^k)$ for all large $k$. It is expected that the balanced metrics converge to a constant scalar curvature metric as $k$ 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