On the virtual invariants of zero entropy groups of compact Kähler manifolds

Let $X$ be a compact Kähler manifold. We study subgroups $G\le \mathrm{A}\mathrm{u}\mathrm{t}(X)$ of biholomorphic automorphisms of zero entropy when ${\mathrm{A}\mathrm{u}\mathrm{t}}^0(X)$ is compact (e.g. when ${\mathrm{A}\mathrm{u}\mathrm{t}}^0(X)$ is trivial). We show that the virtual derived length ${\mathrm{\ell }}_{\mathrm{v}\mathrm{i}\mathrm{r}}(G)$ of $G$ satisfies ${\mathrm{\ell }}_{\mathrm{v}\mathrm{i}\mathrm{r}}(G)\le \dim X-\kappa (X)$, where $\kappa (X)$ is the Kodaira dimension of $X$. Modulo the main conjecture of our previous work concerning the essential nilpotency class, we obtain the same upper bound $c_{\mathrm{v}\mathrm{i}\mathrm{r}}(G)\le \dim X-\kappa (X)$ for the virtual nilpotency class $c_{\mathrm{v}\mathrm{i}\mathrm{r}}(G)$, together with a geometric description of the $G$-action on $X$ when the equality holds.