# References on quaternionic geometry

Is there any analog, in the quaternionic setting, of Kahler potentials?

In particular I'm interested in a similar construction of the Fubini-Study 2-form on $\mathbb{P}^1(\mathbb{C})$ over the quaternionic projective line $\mathbb{P}^1 (\mathbb{H}).$ In the complex setting it is well known that $\omega_{\textrm{FS}}=\frac{i}{2} \partial \overline{\partial} \log (|z|^2).$ Is there a natural extension of this form and of this potential over $\mathbb{P}^1 (\mathbb{H})$?

That is, can the standard 4-form $\Omega=\omega_i \wedge \omega_i + \omega_j \wedge \omega_j + \omega_k \wedge \omega_k$ on $\mathbb{P}^1(\mathbb{H})$ be written as some differential operator applied on some function (i.e. the potential)?

This is only a partial answer. On hypercomplex manifolds there is a notion of HyperKahler with Torsion (HKT) metrics. Such metrics are in a sense quaternionic analogues of Kahler metrics. Locally they can be presented as an application of quaternionic Hessian to a smooth function (potential), as essentially was shown in http://arxiv.org/abs/math/0402366 (see also http://arxiv.org/abs/math/0510140 for some further developments, in particular Proposition 8.2 there).

However the quaternionic projective spaces are not hypercomplex manifolds and are not covered by the above mentioned papers.

• Please do not link to the PDFs but to the abstracts. – José Figueroa-O'Farrill Jul 15 '15 at 13:53
• @JoséFigueroa-O'Farrill: Corrected. – MKO Jul 16 '15 at 4:41
• Thank you! It's much friendlier this way to people on slow/roaming connections. – José Figueroa-O'Farrill Jul 16 '15 at 12:20

Actually, a more direct answer to the original question explains why there is nothing like Kähler potential theory in the hypercomplex and quaternionic cases.

Recall what a Kähler potential does: You start with a complex $n$-manifold, i.e., a $2n$-manifold $M^{2n}$ endowed with a torsion-free $\mathrm{GL}(n,\mathbb{C})$-structure $B$, represented by an integrable complex structure tensor $J:TM\to TM$. Now, $\mathrm{U}(n)$ is a maximal compact in $\mathrm{GL}(n,\mathbb{C})$, and we can select a torsion-free $\mathrm{U}(n)$-structure $B'\subset B$ over an open set $U\subset M$ by choosing a smooth function $u\in C^\infty(U)$ such that $\omega = i\partial\bar\partial u$ is a positive $(1,1)$-form. Locally, this is how all of the torsion-free (i.e., Kähler) structures subordinate to the $\mathrm{GL}(n,\mathbb{C})$-structure $B$ are described. Of course, every torsion-free $\mathrm{U}(n)$-structure $B'$ on $M$ underlies a unique torsion-free $\mathrm{GL}(n,\mathbb{C})$-structure $B = B'{\cdot}\mathrm{GL}(n,\mathbb{C})$. Now, the nice thing in this case is that, modulo diffeomorphism, locally at least, integrable $\mathrm{GL}(n,\mathbb{C})$-structures are all the same, i.e., they have no local geometry, whereas the torsion-free $\mathrm{U}(n)$-structures depend locally on $1$ function of $2n$-variables (which is represented exactly by the Kähler potential as above).

Consider what you would need to be true in either the hypercomplex case or the quaternionic case in order to have such an analogue:

First, consider the hypercomplex case: Now, the group is $\mathrm{GL}(n,\mathbb{H})\subset \mathrm{GL}(4n,\mathbb{R})$, with maximal compact $\mathrm{Sp}(n)\subset \mathrm{GL}(n,\mathbb{H})$. Thus, on a $4n$-manifold $M^{4n}$ you'd want a way to start with a torsion-free $\mathrm{GL}(n,\mathbb{H})$-structure $B$ on $M$ (i.e., a hypercomplex structure) and specify the torsion-free $\mathrm{Sp}(n)$-structures (i.e., hyperKähler structures) that it contains. As in the complex/Kähler case, the reverse operation is automatic: Each torsion-free $\mathrm{Sp}(n)$-structure $B'$ lies in a unique torsion-free $\mathrm{GL}(n,\mathbb{H})$-structure $B= B'{\cdot}\mathrm{GL}(n,\mathbb{H})$. However, you immediately see the problem: Modulo diffeomorphism, the torsion-free $\mathrm{GL}(n,\mathbb{H})$-structures in dimension $4n$ depend on $4n^2$ functions of $2n{+}1$ variables, while the the torsion-free $\mathrm{Sp}(n)$-structures in dimension $4n$ depend on only $2n$ functions of $2n{+}1$ variables, so the 'generic' torsion-free $\mathrm{GL}(n,\mathbb{H})$-structure in this dimension does not contain any torsion-free $\mathrm{Sp}(n)$-structures, so there can't be a potential theory like the complex/Kähler potential theory in this case. You might think that you could get around this by considering the underlying $\mathrm{SL}(n,\mathbb{H})$-structures instead, but, modulo diffeomorphism, the torsion-free $\mathrm{SL}(n,\mathbb{H})$-structures in dimension $4n$ depend on $4n^2{-}2n$ functions of $2n{+}1$ variables, which is still too high (except when $n=1$, when $\mathrm{SL}(1,\mathbb{H}) = \mathrm{Sp}(1)$, so there is nothing to do). (This is why they punt and go for HKT structures instead in this case.)

Second, in the quaternionic case, the group is the group is $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)\subset \mathrm{GL}(4n,\mathbb{R})$, with maximal compact $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)\subset \mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$. Thus, on a $4n$-manifold $M^{4n}$ you'd want a way to start with a torsion-free $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$-structure $B$ on $M$ (i.e., a quaternionic structure) and specify the torsion-free $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)$-structures (i.e., quaternion-Kähler structures) that it contains. As in the complex/Kähler case, the reverse operation is automatic: Each torsion-free $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)$-structure $B'$ lies in a unique torsion-free $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$-structure $B= B'{\cdot}\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$. However, again, there's a problem: Modulo diffeomorphism, the torsion-free $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$-structures in dimension $4n$ depend on $2n(2n{+}1)$ functions of $2n{+}1$ variables, while the the torsion-free $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)$-structures in dimension $4n$ depend on only $2n$ functions of $2n{+}1$ variables, so, again, the 'generic' torsion-free $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)$-structure in this dimension does not contain any torsion-free $\mathrm{Sp}(n){\cdot}\mathrm{Sp}(1)$-structures, so, again, there can't be a potential theory like the complex/Kähler potential theory in this case.

For these function counts for the generality of torsion-free structures modulo diffeomorphism, in case you are curious, you might consult my paper: R. Bryant - Classical, exceptional, and exotic holonomies: a status report, in Actes de la Table Ronde de Géométrie Différentielle en l’Honneur de Marcel Berger, Soc. Math. France, 1996, 93–166.

• The argument is nice, but I am not agree that it implies that "there is nothing like Kähler potential theory in the hypercomplex" case; this strongly depends on your interpretation of the argument and eventually on your taste. If I understand correctly, torsion free $Sp(n)$-structures correspond to hyperkahler metrics. However there exists a broader class of metrics, called hyperkahler with torsion (HKT), which do correspond locally to functions (potentials) via application of some quaternionic Hessian. In other words torsion is allowed sometimes. – MKO Jul 24 '15 at 13:42
• Moreover potentials of HKT-metrics are precisely plurisubharmonic functions in the quaternionic sense. Quaternionic pluripotential theory does exist (though it is quite young) as well as theory of quaternionic Monge-Ampere equations, see e.g. arxiv.org/abs/1108.5910 – MKO Jul 24 '15 at 13:48
• @MKO: Yes, it depends on how one interprets 'nothing like'. I agree that, if one is willing to go outside the realm of torsion-free structures (where the complex/Kähler story takes place), there will be analogs in which one can characterize $G$-structures with specified conditions on intrinsic torsion, but I was considering (and I think the OP wanted) something like the story of Kähler potential in the torsion-free case. I should have pointed out, though, that, torsion-free potentials, in an appropriate sense, can exist in hypercomplex and quaternionic cases; it's just that they usually don't. – Robert Bryant Jul 24 '15 at 14:42
• Torsion free potentials on a hypercomplex manifold are precisely those satisfying some specific linear differential equation of third order. You are right, there are usually few (or no) solutions. – MKO Jul 24 '15 at 15:16