On the Yau-Tian-Donaldson conjecture for weighted cscK metrics

Abstract: We establish a version, formulated in terms of non-Archimedean pluripotential theory, of the Yau-Tian-Donaldson conjecture for constant scalar curvature and, more generally, weighted extremal K\"ahler metrics with prescribed compact symmetry group on an arbitrary polarized projective manifold. This is accomplished by extending important previous work of Chi Li to the weighted case, and by establishing general slope formulas for the relevant weighted entropy and energy functionals. Among other things, our approach relies on key a priori estimates for cscK metrics due to Chen-Cheng (recently extended to the weighted case by Di Nezza-Jubert-Lahdili and Han-Liu), and on a crucial estimate for psh envelopes due to Berman-Demailly-Di Nezza-Trapani.