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My question here is going to be this -- but I'll give a bit of background to explain myself in a moment:

What has been done/what results are available on Calabi-Yau cohomology in degree $n \geq 3$ (in particular $n =3$)?

Here by Calabi-Yau cohomology I mean complex oriented cohomology theory with formal group being (equivalent to) the Artin-Mazur formal group $\Phi^n_X$ of a given Calabi-Yau variety of dimension $n$. (So in particular this here is not about cohomology of Calabi-Yau varieties. I am aware that this will now make 90% of all readers frown, but I can't help it.)

To put this in perspective:

In discussion of elliptic cohomology it is traditional to think of the formal group associated with the elliptic spectrum as "being" the formal completion of the given elliptic curve. But the story of equivariant elliptic cohomology shows that more properly that formal group is identified with the formal Picard group of the elliptic curve, namely the perturbation/deformation theory of the moduli of line bundles over it. It just so happens that elliptic curves are self-dual abelian varieties so that both these perspectives are equivalent, but the latter is the fundamental one that generalizes.

In the next step, K3-cohomology is complex oriented cohomology theory with formal group being the formal Brauer group of a K3 surface. This has been discussed.

But there is an obvious continuation of this story to higher $n$, and I am looking for whatever results exist for $n \geq 3$. My understanding is that Michael Hopkins talked about this "Calabi-Yau cohomology" at the Midwest Topology Seminar in 1992, but I haven't seen anything except this pointer.

Notice that the case $n = 3$ is quite compelling from the point of view of physics: while equivariant elliptic cohomology ("CY1-cohomology") is essentially the theory of the modular functor of 3d Chern-Simons theory/2d Wess-Zumino-Witten theory, so the Artin-Mazur formal group $\Phi^3_X$ is just the formal approximation to the intermediate Jacobian which is the phase space of $U(1)$-7d Chern-Simons theory. It should be quite interesting to ask for a (equivariant) CY3-cohomology theory here which similarly captures the geometric quantization of this and hence yields the modular functor for the infamous 6d theory...

There seems to be fairly strong motivation on the physics side for looking at CY3-cohomology, also if one looks at it from the perspective of F'-theory, and as such hypothetical Calabi-Yau cohomology was highlighted at the end of Sati 05.

But so the question is: what is actually already known about CY3-cohomology? For instance: what are sufficient conditions for the Artin-Mazur formal group $\Phi^3_{CY3}$ to be Landweber exact??

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    $\begingroup$ A naive question: I chased your link to the nforum page on Artin-Mazur fg. Looking at it, I don't see what is special about Calabi-Yau varieties, over arbitrary ones of dim n. I'm guessing the CY condition is what's need to make the AMfg one dimensional, yes? $\endgroup$
    – Charles Rezk
    Commented May 18, 2014 at 1:37
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    $\begingroup$ To guarantee that $\Phi^n$ be prorepresentable by a formal group, it suffices to have $\Phi^{n-1}$ be formally smooth. The condition that $H^{n-1}(X, \mathcal{O}_X)=0$ is an overly strong way to guarantee that. Then yes, $\dim H^n(X, \mathcal{O}_X)$ gives the dimension of the formal group $\Phi^n$, so for a Calabi-Yau it is one-dimensional. $\endgroup$
    – Matt
    Commented May 18, 2014 at 1:45
  • $\begingroup$ @CharlesRezk, true, I should have mentioned this. For completeness I have now added a brief remark on that to the relevant nLab entry: ncatlab.org/nlab/show/… (just as Matt says, of course). $\endgroup$
    – Urs Schreiber
    Commented May 18, 2014 at 19:38
  • $\begingroup$ can I ask where the term "calabi-yau cohomology" was first coined? $\endgroup$
    – bananastack
    Commented May 19, 2014 at 14:27
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    $\begingroup$ @Emily thanks for the pointer. Have added it to the nLab here ncatlab.org/nlab/show/K3-spectrum#Nogami10 with discussion here: nforum.ncatlab.org/discussion/5946/k3spectrum/… $\endgroup$
    – Urs Schreiber
    Commented Nov 24, 2020 at 9:31

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