What is the relationship between integrable systems and toric degenerations? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:54:41Z http://mathoverflow.net/feeds/question/2203 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2203/what-is-the-relationship-between-integrable-systems-and-toric-degenerations What is the relationship between integrable systems and toric degenerations? David Treumann 2009-10-23T22:31:20Z 2012-06-16T14:22:24Z <p>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?</p> <p>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.)</p> <p>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> <p>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."</p> http://mathoverflow.net/questions/2203/what-is-the-relationship-between-integrable-systems-and-toric-degenerations/12803#12803 Answer by Allen Knutson for What is the relationship between integrable systems and toric degenerations? Allen Knutson 2010-01-24T01:17:14Z 2010-01-24T01:17:14Z <p>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.</p> <p>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.</p> <p>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).</p> <p>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.</p> <p>Now you have an integrable system on $F_1$, stolen from $F_0$!</p> <p>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 <em>continuous</em> function $F_1 \to F_0$. </p> <p>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 $F_0$.</p> <p>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.</p> <p>Example:</p> <p>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 $0$. </p> <p>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. </p> http://mathoverflow.net/questions/2203/what-is-the-relationship-between-integrable-systems-and-toric-degenerations/99793#99793 Answer by Kiumars for What is the relationship between integrable systems and toric degenerations? Kiumars 2012-06-16T14:22:24Z 2012-06-16T14:22:24Z <p>Dear David, you may find the article "Integrable systems, toric degenerations and Okounkov bodies", arXiv:1205.5249, useful. </p>