On a conjucture of Kashiwara relating Chern and Euler classes of $\mathcal{O}$-modules

Abstract

In this note we prove a conjecture of Kashiwara, which states that the Euler class of a coherent analytic sheaf $\mathcal{F}$ on a complex manifold $X$ is the product of the Chern character of $\mathcal{F}$ with the Todd class of $X$. As a corollary, we obtain a functorial proof of the Grothendieck–Riemann–Roch theorem in Hodge cohomology for complex manifolds.