Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb {C}$ equipped with a very ample Hermitian line bundle $\mathcal {L}$. In the first part of the paper, we show that if there exists a toric degeneration of $X$ satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from $X$ to the special fiber $X_0$ which is a symplectomorphism on an open dense subset $U$. From this we are then able to construct a completely integrable system on $X$ in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions $\{H_1, \ldots , H_n\}$ on $X$ which are continuous on all of $X$, smooth on an open dense subset $U$ of $X$, and pairwise Poisson-commute on $U$. Moreover, our integrable system in fact generates a Hamiltonian torus action on $U$. In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the ‘moment map’ $\mu = (H_1, \ldots , H_n): X \rightarrow \mathbb {R}^n$ is precisely the Newton-Okounkov body $\Delta = \Delta (R, v)$ associated to the homogeneous coordinate ring $R$ of $X$, and an appropriate choice of a valuation $v$ on $R$. Our main technical tools come from algebraic geometry, differential (Kahler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Łojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties $X$, this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.

Integrable systems, toric degenerations and Okounkov bodies

A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory (GIT) quotients come with a natural ample line bundle and hence often a natural projective embedding. This question translates to determining the equations of the moduli space under this embedding. This article deals with one of the most classical quotients, the space of ordered points on the projective line. We show that under any weighting of the points, this quotient is cut out (scheme-theoretically) by a particularly simple set of quadric relations, with the single exception of the Segre cubic threefold, the space of six points with equal weight. We also show that the ideal of relations is generated in degree at most four, and we give an explicit description of the generators. If all the weights are even (e.g., in the case of equal weight for odd n), then we show that the ideal of relations is generated by quadrics

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The equations for the moduli space of $n$ points on the line

Let X be a smooth complex projective variety of dimension n equipped with a very ample Hermitian line bundle L. In the first part of the paper, we show that if there exists a toric degeneration of X satisfying some natural hypotheses (which are satisfied in many settings), then there exists a completely integrable system on X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions H_1, ... H_n on X which are continuous on all of X, smooth on an open dense subset U of X, and pairwise Poisson-commute on U. Moreover, we show that in many cases, we can construct the integrable system so that the functions H_1, ..., H_n generate a Hamiltonian torus action on U. In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the "moment map" \mu = (H_1, ..., H_n): X to R^n is precisely the Okounkov body \Delta = \Delta(R, v) associated to the homogeneous coordinate ring R of X, and an appropriate choice of a valuation v on R. Our main technical tools come from algebraic geometry, differential (Kaehler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Lojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties X, this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.

Speyer and Sturmfels (SpSt) associated Grobner toric degenerations Gr2(C n ) T of Gr2(C n ) to each trivalent tree T with n leaves. These degenerations induce toric degenerations M T r of Mr, the space of n ordered, weighted (by r) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers as stratified symplectic spaces and describe the action of the compact part of the torus as "bendings of polygons." We prove the conjecture of Foth and Hu (FH) that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida (KY).

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The Toric Geometry of Triangulated Polygons in Euclidean Space

Matroids and Geometric Invariant Theory of torus actions on flag spaces

We show that a weight variety, which is a quotient of a flag variety by the maximal torus, admits a flat degeneration to a toric variety. In particular, we show that the moduli spaces of spatial polygons degenerate to polarized toric varieties with the moment polytopes defined by the lengths of their diagonals. We extend these results to more general Flaschka-Millson hamiltonians on the quotients of products of projective spaces. We also study boundary toric divisors and certain real loci.

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Yi Hu

Papers

Toric degenerations of weight varieties and applications