Integrable systems, toric degenerations and Okounkov bodies

Abstract

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 (Kähler) 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.