Stability of weighted extremal manifolds through blowups

Abstract: In a previous paper, we showed that the blowup of a weighted extremal K\"ahler manifold at a relatively stable fixed point admits a weighted extremal metric. Using this result, we prove that a weighted extremal manifold is relatively weighted K-polystable. In particular, a weighted cscK manifold is weighted K-polystable. This strengthens both the weighted K-semistability proved by Lahdili and Inoue, and the weighted K-polystability with respect to smooth degenerations by Apostolov--Jubert--Lahdili, allowing for possibly singular degenerations.