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.