# Is the mirror of a noncompact hyperkaehler manifold also hyperkaehler?

This is essentially a follow-up question from 'Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?'. Verbitsky's theorem in (https://arxiv.org/pdf/hep-th/9512195.pdf) says that certain compact hyperkaehler manifolds are their own mirrors. What about noncompact hyperkaehler manifolds?

Are the mirrors of noncompact hyperkaehler manifolds also hyperkaehler? Are there at least some examples where this is true? I think at least for noncompact K3 manifolds, this should be known.

In particular, I am interested in noncompact hyperkaehler manifolds which are cotangent bundles. For example, the cotangent bundles of hermitian symmetric spaces are hyperkaehler, as discussed in 'Is the cotangent bundle to a Kahler manifold hyperkahler?'

• First of all one should be wary of the idea/slogan that mirror symmetry for hyperkahler manifolds is related to hyperkahler rotation: see Gross's answer to mathoverflow.net/questions/119899/… – Daniel Pomerleano Jun 12 '17 at 19:57
• Second, for general noncompact hyperkahler varieties and for cotangent bundles in particular it seems likely that a mirror will not be a variety at all but a Landau-Ginzburg pair $(X,W)$. The case $T^*\mathbb{C}P^1$ is worked out in Section 9.2. of arxiv.org/pdf/1205.0053.pdf. This can be generalized to e.g. $T^*\mathbb{C}P^n$ although to my knowledge the resulting mirror construction does not appear in the literature. Finally, there are indeed instances of log Calabi-Yau varieties which are self-mirror in a suitable sense. You can find such instances in arxiv.org/abs/1106.4977 – Daniel Pomerleano Jun 12 '17 at 20:03
• That seems strange, if $T^*\mathbb{C}P^1$ is Hyperkaehler, then it is Calabi-Yau, and it should have a mirror Calabi-Yau without any need for a Landau-Ginzburg superpotential $W$. – Mtheorist Jun 13 '17 at 3:41
• In the non-compact case and speaking heuristically, it is generally appreciated that it is log CY varieties that "should" have geometric mirrors(i.e. W=0). To see at an intuitive level what happens on a general non-compact CY, one obvious thing to keep in mind is that from the SYZ perspective one wants at the very least conditions that ensure that the holomorphic volume form is not exact. Otherwise there cannot be any special lagrangian tori. On a general noncompact CY this won't be the case... I recommend consulting the references mentioned above for more info and some precise theorems. – Daniel Pomerleano Jun 13 '17 at 19:04