Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows:

A) Is there a classification of the possible homeomorphism types of such $M$?

B) Is there a classification of the possible diffeomorphism types of such $M$?

Hitchin has proved that if $\vert \tau(M) \vert = \frac{2}{3}\chi (M)$ then $(M,g)$ is isometric to a K3 surface equipped with a Ricci flat metric. Here $\tau(M)$ denotes the signature of $M$ and $\chi(M)$ denotes its Euler characteristic. Therefore, through my two questions above I am wondering about the diffeo or homeo type of $M$ in the case $2\chi (M) + 3 \tau(M) >0$.

Alternatively, if $(M,g)$ is irreducible, simply connected and Ricci-flat four-manifold, Berger classification implies that its holonomy group must be either SU(2) or SO(4). If Hol($g$) = SU(2) then $(M,g)$ is a K3 surface equipped with a Calabi-Yau metric. Otherwise $(M,g)$ has generic holonomy. It is currently not known if there exist Ricci-flat compact manifolds of generic holonomy. In this context, questions A) and B) about concern the possible diffeo or homeo types of simply connected compact Ricci-flat four manifolds of generic holonomy. The previous remarks suggests a third question:

C) Is every Ricci flat metric on a K3 surface Calabi-Yau?


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    $\begingroup$ My understanding is that all your questions are open. $\endgroup$ – Igor Belegradek Oct 19 '18 at 1:47
  • $\begingroup$ @IgorBelegradek I am surprised. It is not possible to even classify the homeomorphism type of compact four-manifolds carrying Ricci flat metrics? $\endgroup$ – Bilateral Oct 19 '18 at 15:39
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    $\begingroup$ Why is it surprising? As you say very little is known about Ricci-flat 4-manifolds with $SO(4)$ holonomy, and a classification would probably require understanding such manifolds. $\endgroup$ – Igor Belegradek Oct 20 '18 at 1:36
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    $\begingroup$ Actually, the answer to C) is known: any Ricci-flat metric on a closed smooth manifold homotopy equivalent to a K3 surface is hyperkahler (and hence Calabi–Yau). See A. L. Besse "Einstein manifolds", Theorem 6.40. About your other questions: Does the 4-sphere admit a Ricci-flat metric? This is the kind of issues one has to tackle. $\endgroup$ – Igor Belegradek Oct 20 '18 at 2:00
  • $\begingroup$ @IgorBelegradek thanks for the answers. I was a bit surprised because although no example of Ricci flat compact manifold of generic holonomy is known, I thought that assuming the existence of such metric could be used to work out obstructions and perhaps give a list of possible homeo types. $\endgroup$ – Bilateral Oct 20 '18 at 12:50

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