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.

  • $\begingroup$ In GIT, stabilaisers of stable points are reductive. There are results along these lines for csck and extremal metrics that go back to Lichnerowicz and Calabi. For general type the automorphism group is discrete. And holomorphic vector fields are parallel wrt Calabi Yau metrics, in particular there are no non-trivial if the holonomy is SU(n) or Sp(n). Up to a covering the Bogomolov thm says that you are a product of these with flat tori. Regarding the algebraic proof in the Fano case, I know it was open two years ago. I don't know now. $\endgroup$ – Martin de Borbon Jul 29 '20 at 8:40
  • $\begingroup$ What do you mean by an algebraic proof? Kähler-Einstein is not an algebraic property. $\endgroup$ – abx Jul 29 '20 at 9:03
  • $\begingroup$ I guess it means K-polystable Fano $\endgroup$ – Martin de Borbon Jul 29 '20 at 9:42
  • 2
    $\begingroup$ Then this paper is probably relevant. $\endgroup$ – abx Jul 29 '20 at 13:50

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