Résumé
In K-stability, the delta invariant of a Fano variety encodes the existence
of Kähler-Einstein metrics. We introduce a weighted analytic delta invariant,
and a reduced version, that characterize the existence of weighted solitons. We
further prove a sufficient condition of existence of weighted cscK metrics in
terms of this invariant. We elucidate the relation between the weighted delta
invariant and the greatest lower bound on the weighted Ricci curvature, called
the weighted beta invariant. We provide a general upper bound for the weighted
beta invariant in terms of moment images. Finally, we investigate how the
geometry of semisimple principal fibrations, whose basis is not assumed to be
cscK, allows to estimate their beta invariant in terms of the basis and the
weighted fiber. Most of our statements are new even in the trivial weights
settings, that is, for Kähler-Einstein and cscK metrics.