The Yau–Tian–Donaldson conjecture for K-stable maps

Let $p:(X,L)\to(Y,T)$ be a map of polarized varieties, and suppose that $p$ has discrete automorphism group; this occurs, for instance, if $p^*T$ is ample. Let $\alpha\in c_1(T)$ be a Kähler metric, and let $c_1(L)$ denote the polarization class on $X$. A $(1,1)$-form $\omega\in c_1(L)$ is $p^*\alpha$-twisted constant scalar curvature Kähler if it is positive and satisfies

$$\operatorname{Scal}(\omega)-\Lambda_{\omega}p^*\alpha\equiv \operatorname{const}.$$

The Yau–Tian–Donaldson conjecture for maps. The class $c_1(L)$ admits a $p^*\alpha$-twisted cscK metric if and only if the map $p:(X,L)\to(Y,T)$ is uniformly K-stable. This is the proposed analogue for maps of the Yau–Tian–Donaldson conjecture relating canonical Kähler metrics to K-stability. Its status is not resolved in the supplied source context.