Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I am looking for examples of Hamiltonian polar isometric actions of a compact Lie group on a Kahler-Einstein (or perhaps just Kahler) manifold, that admits a Lagrangian section.

Recall that an isometric action on $M$ is polar if there exists a submanifold $\Sigma\subset M$, called section, that meets all orbits orthogonally. Any section is automatically a totally geodesic submanifold, and there may be many different sections on $M$. The examples I am looking for should have a section $\Sigma\subset M$ that is also a Lagrangian submanifold, i.e., $\omega|_\Sigma=0$ and $\dim M=2\dim \Sigma$.

So far, the only example I can see is the standard torus action of $T^n$ on $\mathbb C P^n$, with the Fubini metric.

This action is Hamiltonian and isometric. It is also polar (in fact, by a Theorem of Podesta-Thorbergsson, if a torus $T^n$ acts on a compact Kahler manifold $M^{2n}$ of complex dimension $n$ and positive Euler characteristic, then this action is polar). Finally, the usual totally real embedding of $\mathbb R P^n$ into $\mathbb C P^n$ gives a Lagrangian section for the action.


With such strong hypothesis I would imagine that such actions are perhaps even classified, but I could not find anything in the literature about Lagrangian sections. Most classification results have to do with coisotropic actions, where it is required that the orbits satisfy $\omega|_{G(x)}=0$; while I am interested in an "orthogonal" version of that, i.e., I want the section to be coisotropic (even more, Lagrangian).

share|improve this question
add comment

1 Answer

I don't know what the global classification might be, but examining the structure equations for such a structure shows that the 'local' classification is reasonable.

Let $(M,g,\omega)$ be a Kähler manifold (of real dimension $2n$) with a symplectic and isometric polar action $G\times M\to M$ and let $\Sigma^n\subset M$ be a Lagrangian section. (Since the consideration will be local, this can be just a local section for now.) Because the section has dimension $n$, the generic orbits (which meet $\Sigma$ orthogonally) must be of (real) codimension $n$ and must, themselves be Lagrangian. Let us restrict our attention to the open subset of $M$ consisting of $n$-dimensional $G$-orbits.

As the OP observes, $\Sigma$ must be totally geodesic. If $m\in\Sigma$ is fixed and $G_m\subset G$ is the stabilizer of $m$, then $G_m$, which preserves the $n$-dimensional orbit $G\cdot m$ and hence its tangent space, must preserve the orthogonal to this tangent space, i.e., $T_m\Sigma$. Hence, $G_m$ must preserve $\Sigma$ since the image of $\Sigma$ under an element of $G_m$ must be a totally geodesic Lagrangian submanifold that is tangent to $\Sigma$ at $m$. Thus the images of $\Sigma$ under $G$ define a Lagrangian foliation of $M$ that is transverse to the Lagrangian foliation defined by the orbits of $G$.

Following along this kind of argument, one sees that, locally, one can choose local holomorphic coordinates $z = (z^i) = x + i\ y$ so that a Kähler potential $\phi$ for $\omega$ can be found that is a function of $x$ alone and the action of $G$ is just translation in the $y$-directions. In particular, $G$ is abelian, and, in order to make the group compact, one has to take $y\in \mathbb{R}^n$ to be defined modulo a lattice $\Lambda\subset\mathbb{R}^n$. Locally, the section $\Sigma$ can be taken to be defined by $y=0$.

In order for the potential $\phi = \phi(x)$ to define a Kähler metric, $\phi$ has to be convex relative to the affine structure on $\mathbb{R}^n$. Then, in order for this to define a Kähler-Einstein metric, $\phi$ has to satisfy a Monge-Ampère equation.

I believe that there is a section in a Chapter of Besse's Einstein manifolds that describes these metrics (which, I think, were first seriously investigated by Calabi). However, I don't have a copy of the book here at home with me this weekend.

Note that this is just a local classification on a dense open set. Of course, to deal with the case of $M$ compact, one is dealing with the case of toric Kähler manifolds, but I'm not sure which toric Kähler manifolds admit a polar section. However, I would imagine that this must be well-studied; one should look for more information in the toric literature. In particular, I recommend two papers (out of many possibilities):

V. Guillemin, Kähler metrics on toric varieties, J. Diff. Geom. 40 (1994), 285–309. MR 95h:32029

and

M. Abreu, Kähler geometry of toric varieties and extremal metrics, Internat. J. Math. 9 (1998), 641–651. MR 99j:58047

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.