What is the relationship between integrable systems and toric degenerations?

Question

Given an integrable system on a Kahler manifold X, is there a way to associate a toric degeneration of X whose Milnor fibers are related to the fibers of the integrable system?

An integrable system is, at least, a map from X to R^n whose coordinate functions Poisson commute. The moment map of a Hamiltonian torus action will have this property, but there are other examples. For instance, the flag variety GL(n,C)/B has a famous integrable system and a famous toric degeneration, both of which are related to the same polytope--a Gelfand-Tsetlin polytope. (Famous but I don't know the original references for these constructions.)

Given a toric degeneration Y --> C, you can try to construct an integrable system on a general fiber Y1 by flowing along a gradient vector field from Y1 to Y0 (the special fiber, a toric variety) and projecting to R^n via the moment map of the torus action on Y0. I heard that this doesn't work on the nose, but that it does work well enough that you can at suitable points identify the fibers of e.g. the Gelfand-Tsetlin integrable system with the Milnor fibers of the Gelfand-Tsetlin toric degeneration. Possibly starting with an integrable system and trying to construct a toric degeneration is easier and more algebraic.

P.S. Some references after all: Guillemin and Sternberg, "The Gelfand-Cetlin system and quantization of the complex flag manifolds," and Gonciulea and Lakshmibai, "Degenerations of Flag and Schubert varieties to toric varieties."

Answer 1

I very nearly wrote my PhD thesis on this topic. Here's as much as I was able to figure out, though it's hardly a direct answer to your question.

1) Say your total space is K\"ahler, and your fibers are compact. Then you can define a Levi-Civita connection on any open set consisting of smooth fibers. It turns out that this connection generates symplectomorphisms between the fibers.

2) In toric degenerations, the torus acts on the total space of the family, mostly moving them around, but preserving the zero fiber (which is why it's toric).

1+2?) Now imagine you use (1) to give a map from your general fiber $F_1$ to your special fiber $F_0$. Map further, to ${\mathfrak t}^*$, using the moment map on the toric variety.

Now you have an integrable system on $F_1$, stolen from $F_0$!

There's a problem: since $F_0$ isn't smooth, we can't actually use (1) to make the map. The hope is to take limits along the horizontal vector field to define a continuous function $F_1 \to F_0$.

3) It turns out that this is the same as following the gradient flow for the norm square of the moment map. And limits of real-analytic gradient flows on smooth varieties are well-understood, by Lojasiewicz. So if your total space is smooth, you can use this to show that the map $F_1 \to F_0$ is well-defined, continuous, and smooth away from the singularities in $F_0$.

I never got around to investigating how things change if the total space is singular (as in the Gel'fand-Cetlin-Sturmfels-Gonciulea-Lakshmibai degeneration motivating the questioner, and me too). Of course you can pick a resolution of singularities, and I guess you can ask that the metric on the exceptional fibers be very very small, and use that to generalize Lojasiewicz' results. But I never worked on this seriously.

Example:

Let the family be $det : C^{2\times 2} \to C$. Then the $0$ fiber is the cone over $P^1 \times P^1$, so a toric variety, but the fiber over $1$ is $SL(2)$. That has a $T^2$ action, by left and right multiplication by its maximal torus, but doesn't have the rescaling action that the $0$ fiber enjoys. One can actually solve the ODE defined by the Levi-Civita/gradient flow and write down the map $SL(2) \to det^{-1}(0)$. It collapses $SU(2)$ to the singular point $0$.

What is the integrable system? Regard $SL(2)$ as $T^* S^3$, and the action variable as $(p,\vec v) \mapsto |\vec v|$. This generates unit-speed gradient flow on $T^* S^3$, which breaks down at zero vectors (the $SU(2) = S^3$) because they don't know which direction to go.

Answer 2

Dear David, you may find the article "Integrable systems, toric degenerations and Okounkov bodies", arXiv:1205.5249, useful.