The noncommutative residue formula for differential operators on line bundles

Let $M$ be a compact complex manifold, let $obreak\lambda$ be a line bundle on $M$, and let $\boldsymbol1_\lambda$ denote its identity differential operator. Write $\operatorname{Tr}_\lambda(\boldsymbol1_\lambda)$ for the noncommutative residue and $\chi(\lambda)$ for the Euler characteristic of $\lambda$. Noncommutative residue formula.

$$\operatorname{Tr}_\lambda(\boldsymbol1_\lambda)=C(M)\cdot\chi(\lambda),$$

where $C(M)$ does not depend on $\lambda$. The preceding discussion establishes that the residue is an invariant of the line bundle; the displayed proportionality is presented as the paper's conjectural formula, with the dependence on $M$ captured by $C(M)$.