Relative uniform Yau--Tian--Donaldson correspondence for projective bundles over a curve

This paper is concerned with a relative uniform Yau–Tian–Donaldson correspondence, in terms of test configurations, for the projectivization $\mathbb{P}(E)$ of a holomorphic vector bundle $E$ over a smooth curve. For any Kähler class $[\omega ]$ on $\mathbb{P}(E)$, we construct Kähler test configurations, which we call \emph{compatible test configurations}. They are obtained by gluing horospherical test configurations from the fibers, arising from convex functions on a suitable moment polytope $\mathrm{\Delta }$ following the construction of Delcroix, to the principal bundle associated with $\mathbb{P}(E)$. Using the generalized Calabi ansatz of Apostolov–Calderbank–Gauduchon–Tønnesen-Friedman on these test configurations, we show that the relative uniform stability of $(\mathbb{P}(E),[\omega ])$ for compatible test configurations implies the existence of an extremal metric in this class, thereby establishing the equivalence. Along the way, we prove that these two conditions are equivalent to the weighted uniform stability of $\mathrm{\Delta }$ for suitable explicit weight functions defined from the topological data of $\mathbb{P}(E)$.