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I want to know if there exists examples of Calabi-Yau manifolds with $\mathbb{T}^2$-invariant $SU(n)$-structure. In particular these actions are both Killing and holomorphic. I am especially interested in the case when $n\geq4$. I would be happy even with local examples.

A CY $n$-fold (for me) is a Kahler manifold with holonomy group equal to $SU(n)$, in particular I don't include products of lower dimensional CY manifolds. Since CY manifolds are Ricci flat, this implies that these examples (if any) cannot be compact unless they are the products of a CY $(n-1)$-fold and $\mathbb{T}^2$ (elliptic curve). So the question is asking for non-compact examples.

In the case of $S^1$-invariant CY manifolds, I am aware of a few examples; in real dimension $4$ the Eguchi-Hanson and Taub-Nut metrics, and in higher dimensions there are examples on line bundles over CY manifolds and higher dimensional version of "Taub-Nut metrics". I would also be interested in knowing more examples of those as well.

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As far as I understand, such metrics can be obtained by the so-called "Calabi-anszats":

http://www.numdam.org/article/ASENS_1979_4_12_2_269_0.pdf

Namely, one can construct a Calabi-Yau metric on the total space of the canonical bundle of a compact Kahler-Einstein manifold of positive scalar curvature, i.e. canonical bundles of some Fano manifolds. It is well-known, that there exist quite a lot of compact Kahler-Einstein manifolds with a $T^2$-symmetry ($\mathbb CP^n, \mathbb CP^1\times \mathbb CP^1$, some toric varieties). I guess, that Calabi's metric can be constructed so that the symmetries of the base extend to symmetries of the total space.

However, I have not read the paper of Calabi carefully, for this reason I don't know what is the precise theorem in this paper where this is proven. Also, unfortunately, usually when people refer to this paper they never cite the precise result...

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  • $\begingroup$ I was aware of these metrics but it wasn't clear to me that the SU(4)-structure is preserved. I will have a closer look. But as far as I know these examples are only U(1) invariant. I’m not entirely sure how the argument to lift the symmetry would work $\endgroup$ – u184 Dec 19 '19 at 21:52
  • $\begingroup$ Maybe I misunderstand something, but the action should preserve the Calabi-Yau metric, the holomorphic volume form and the holomorphic structure. Does this mean that it preserves the SU(n) structure? (I guess it does) $\endgroup$ – aglearner Dec 19 '19 at 22:41
  • $\begingroup$ Yep, that’s precisely it. Since the holomorphic volume form determines the complex structure, the last condition is automatic. So I believe from Calabi’s paper that since the potential is U(1)-invariant, the CY structure is preserved. $\endgroup$ – u184 Dec 19 '19 at 22:57

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