Let $(X, \omega)$ be a compact Kähler manifold. We will say that $X$ is **Calabi–Yau** if the first Chern class of the anti-canonical bundle is trivial, in symbols: $c_1(-K_X)=0$; we will say $X$ is of **general type** if $c_1(-K_X) <0$; and we will say that $X$ is **Fano** if $c_1(-K_X)>0$. It is well-known that compact Kähler manifolds with $c_1(-K_X) <0$ or $c_1(-K_X)=0$ admit Kähler–Einstein metrics (unique in each Kähler class). For Fano manifolds, however, there are obstructions to the existence of Kähler–Einstein metrics: Matsushima taught us long ago that the **automorphism group of a Kähler–Einstein Fano manifold was necessarily reductive** (i.e., the complexification of a compact Lie group); Futaki gave us a more analytic invariant in the 80's, the so-called **Futaki-invariant** (a Lie algebra character for the Lie algebra of holomorphic vector fields); and finally this culminated into Tian's **K-stability** (an extension of the classical notions of stability due to Mumford).

**Question:** There should be a unified perspective here when it comes to the role of the automorphism group. That is, if a Kähler–Einstein Fano has a reductive group of automorphisms, this *should* be true also for Calabi–Yau manifolds and manifolds of general type. Is what *should* be true, *actually* true, i.e., **is the automorphism group of a Calabi–Yau manifold or a manifold of general type reductive?**

**Remarks on how to prove Matsushima:** My question proposes that the standard method of proof of Matsushima's result is not the most enlightening. Indeed, the proof relies on the Weizenböck/Bochner fomula $$\Delta \eta = \nabla^{\ast} \nabla + \text{Ric}(\eta)$$ to conclude that holomorphic vector fields can be expressed as $$\zeta = df + d^c h,$$ where $f$ and $h$ are eigenfunctions of the Laplacian with eigenvalue $\frac{1}{n} \text{Scal}$. Similarly, the above Bochner formula allows you to prove that the Killing vector fields on a KE Fano can be writte $\xi = J dh$ ($h$ again an eigenfunction with the same eigenvalue). Then the Lie algebra of holomorphic vector fields is the complexification of the Lie algebra of Killing vector fields (whose corresponding Lie group is compact and real).

Returning to what I remarked above, the method of proof, therefore, simply manipulates the sign of the curvature in the Bochner formula.

**Perhaps it would be enlightening to know an algebraic proof of the reductivity of the automorphism group of a KE Fano.**

**Note:** We do not require Calabi–Yau manifolds to be simply connected.

*Apologies in advance if this is well-known.*