Donaldson–Sun conjecture on asymptotic Bergman kernel bounds

Let $n\in\mathbb{N}$, $c,d>0$, and $\eta\in(0,1)$. Let $(M,\omega,\mathcal{L},h)$ be a polarized Kähler manifold with $|\operatorname{Ric}|\leq 1$ and $\operatorname{diam}(M,\omega)\leq d$. Suppose that every metric $r$-ball in $M$ satisfies

$$\operatorname{Vol}(B_r)\geq c\frac{\pi^n}{n!}r^{2n},\qquad \forall r\in(0,d).$$

Let $D=D(c)$ be the least integer such that every integer less than or equal to $c^{-1}$ divides $D$. Donaldson–Sun's conjecture. There is a number $m_0$ such that

$$\eta\leq (mD)^{-n}\rho_{mD}\leq\frac{1}{c\eta},\qquad \forall m>m_0.$$

This conjecture asserts that, under two-sided Ricci curvature, diameter, and non-collapsing volume-ratio bounds, the normalized Bergman kernel is uniformly controlled from below and above for all sufficiently large multiples of $D$.