Remarks on the higher dimensional Suita conjecture

To study the analog of Suita’s conjecture for domains $D \subset \mathbb {C}^n$, $n \geq 2$, Błocki introduced the invariant $F^k_D(z)=K_D(z)\lambda \big (I^k_D(z)\big )$, where $K_D(z)$ is the Bergman kernel of $D$ along the diagonal and $\lambda \big (I^k_D(z)\big )$ is the Lebesgue measure of the Kobayashi indicatrix at the point $z$. In this note, we study the behaviour of $F^k_D(z)$ (and other similar invariants using different metrics) on strongly pseudconvex domains and also compute its limiting behaviour explicitly at certain points of decoupled egg domains in $\mathbb {C}^2$.