Schapira–Schneiders microlocal Euler class conjecture

Let $X$ be a complex manifold of dimension $d$, and let ${\cal M}^\bullet$ be a complex of ${\cal D}_X$-modules with bounded good cohomology. Choose a filtration compatible with the order filtration on ${\cal D}_X$, and let $\sigma({\cal M}^\bullet)$ be its symbol complex. Write $\mu\operatorname{eu}({\cal M}^\bullet)$ for its microlocal Euler class. For a closed conic subvariety $\Lambda\subset T^*X$ containing $\operatorname{char}({\cal M}^\bullet)$, let $ch_\Lambda$ denote the Chern character with supports and let $\pi:T^*X\to X$ be the canonical projection. Schapira–Schneiders' conjecture.

$$\mu\operatorname{eu}({\cal M}^\bullet)=\left[ch_\Lambda(\sigma({\cal M}^\bullet))\smile\pi^*Td(TX)\right]^{2d}.$$

This conjecture identifies the microlocal Euler class of a complex of differential modules with the top-degree component of the supported Chern character of its symbol, multiplied by the Todd class. It is presented as a conjecture of Schapira and Schneiders and is the Riemann–Roch input for the microlocal index formula developed in the paper.