Critical height–moduli height conjecture for endomorphisms of projective space

Let $\mathcal{M}_d^N$ be the moduli space of degree-$d$ endomorphisms of $\mathbb{P}^N$, let $h_{\mathcal{M}}$ be an ample moduli height, and let $\widehat h^{\mathrm{crit}}$ be the critical height defined from the Bogomolov canonical height of the critical divisor. Critical height–moduli height conjecture. There is a Zariski closed subset

$$L_d^N\subsetneq\mathcal{M}_d^N$$

such that, on $\mathcal{M}_d^N(\overline{\mathbb{Q}})\smallsetminus L_d^N$,

$$\widehat h^{\mathrm{crit}}\asymp h_{\mathcal{M}}.$$

Here $A\asymp B$ means that the two heights are commensurate. The conjecture generalizes Ingram’s theorem to higher dimension; its general validity remains open.