# Automorphism group of compact hyperkähler manifolds

Let $$M$$ be a compact simply-connected hyperkähler manifold, and let $$\mathrm{Aut}(M)$$ be the automorphism group of $$M$$, i.e. the group of tri-holomorphic diffeomorphisms preserving the metric.

Then, $$\mathrm{Aut}(M)$$ is a Lie group since the isometry group of a compact Riemannian manifold is a Lie group as well as the group of biholomorphisms of a compact complex manifold.

Questions:

(1) Is $$\mathrm{Aut}(M)$$ discrete, or are there examples where $$\dim \mathrm{Aut}(M)>0$$?

(2) Can $$\mathrm{Aut}(M)$$ contain a positive-dimensional compact Lie group?

• Well, the automorphism group of compact complex tori contains all the translations... – Francesco Polizzi Nov 6 '19 at 19:12
• @FrancescoPolizzi Ah yes, obviously... I should have added simply connected, as this is a standard assumption for compact hyperkähler manifolds. – SHP Nov 6 '19 at 19:23
• I think the argument using Bochner works just like for Calabi Yaus: the Killing fields are dual to harmonic 1-forms, which vanish because it is simply connected. – Ben McKay Nov 6 '19 at 19:25
• I think that even without simple-connectedness, you can say something, provided the holonomy representation is irreducible and $M$ is connected. Ricci-flatness implies that Killing vector fields are parallel, and these vanish by irreducibility. The isometry group must be compact because $M$ is compact, so it must be finite in this case. – Paul Reynolds Nov 6 '19 at 21:09
• @PaulReynolds: there exist K3 surfaces with infinite (discrete) group of automorphisms. – Francesco Polizzi Nov 6 '19 at 22:34

By $$\mathrm{Aut}(X)$$ I denote the group of holomorphic automorphisms of a complex manifold $$X$$. I assume that hyperkähler manifolds are compact and simply-connected.

There are two standard series of deformation types of hyperkähler manifolds:

1. $$K3^{[n]}$$-type. Hyperkähler manifold that is a deformation of a Hilbert scheme of $$n$$-points on a given projective $$K3$$-surface.

2. Kummer $$n$$-type. Hyperkähler manifold that is a deformation of a fiber over zero of the addition morphism $$T^{[n+1]}\rightarrow T$$, where $$T$$ is an abelian surface and $$T^{[n+1]}$$ is its Hilbert scheme of $$n+1$$-points.

For these series you have the following results.

Lemma 7.1.3 of Monagardi's thesis. Let $$X$$ be a hyperkähler of $$K3^{[n]}$$-type. Then the map $$\mathrm{Aut}(X)\rightarrow O\left(\mathrm{H}^2(X,\mathbb{Z})\right)$$ is injective, where the right hand side is the orthogonal group with respect to Beauville-Bogmolov form on $$X$$.

Remark. The case $$n=1$$ of the result above is precisely the case of $$K3$$-surfaces (any $$K3$$ surface is deformation equivalent to a projective $$K3$$-surface). There is even more precise celebrated result of Pyatetskii-Shapiro and Shafarevich.

Theorem by K.Oguiso. Let $$X$$ be a hyperkähler of Kummer $$n$$-type. Then the map $$\mathrm{Aut}(X)\rightarrow \mathrm{Aut}_{\mathbb{Z}}\left(\mathrm{H}^*(X,\mathbb{Z})\right)$$ has trivial kernel.

Since every compact smooth manifold is homeomorphic to a finite CW-complex (by virtue of Morse theory), we derive that the ring $$\mathrm{H}^*(X,\mathbb{Z})$$ is finitely generated as an abelian group. Thus for these two standard series groups of automorphisms are discrete.

I strongly recommend Mongardi's thesis as an excellent source of various results and references in this direction.

Now the results concerning the two series above are important due to the following remark by Huybrechts (page 4 of this paper).

Let us state explicitly the following immediate consequence of (2.1):

2.2.Two birational hyperkähler manifolds are deformation equivalent.

In particular, their Hodge, Betti, and Chern numbers coincide. The result was used to show that most of the known examples, with the exception of O’Grady’s exceptional examples in dimension six and ten, are deformations of the two standard series provided by Hilbert schemes of points on K3 surfaces and generalized Kummer varieties.

You may also be interested in Huybrechts exhibition of a Torelli theorem due to M.Verbitsky for HK manifolds, which is also related to your questions.

As for your questions I don't know the answer, but the results presented above and Huybrechts remark suggest that if there are continuous subgroups, then you should look for them in the realm of O'Grady's examples.