Donaldson's twisted cscK conjecture

Let $M$ be a compact Kähler manifold with Kähler class $[\omega]$, let $\chi$ be a closed positive $(1,1)$-form, and let $t\in[0,1]$. A metric solving the corresponding twisted constant scalar curvature Kähler equation is a twisted cscK metric, and its algebro-geometric stability condition is called twisted K-stability. Twisted cscK conjecture. There exists a twisted cscK metric if and only if it is twisted K-stable. This is presented as the twisted analogue of the Yau–Tian–Donaldson conjecture; the general equivalence is open.