Arveson–Douglas Schatten-class conjecture

Let $I \subset \mathbb{C}[z_1,\ldots,z_d]$ be a homogeneous ideal of infinite co-dimension, let

$$V(I)=\{z\in\mathbb{C}^d:p(z)=0\text{ for all }p\in I\},$$

and set $I^\bot=H^2_d\ominus I$. Define $S^I=(S_1,\ldots,S_d)$ on $I^\bot$ by

$$S_i=P_{I^\bot}M_{z_i}\big|_{I^\bot}.$$

The commutator of two operators is in the Schatten class $\mathcal{S}^p$ when it belongs to that Schatten ideal. Arveson–Douglas conjecture. The commutators

$$S_jS_k^*-S_k^*S_j$$

belong to the Schatten class $\mathcal{S}^p$ for all $p>\dim V(I)$ and all $1\leq j,k\leq d$. The conjecture refines essential normality by prescribing Schatten-class summability of the commutators; it remains open in general, but has been verified in special cases.