Integrable systems, toric degenerations and Okounkov bodies

Let $$X$$X be a smooth projective variety of dimension $$n$$n over $$\mathbb {C}$$C equipped with a very ample Hermitian line bundle $$\mathcal {L}$$L. In the first part of the paper, we show that if there exists a toric degeneration of $$X$$X satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from $$X$$X to the special fiber $$X_0$$X0 which is a symplectomorphism on an open dense subset $$U$$U. From this we are then able to construct a completely integrable system on $$X$$X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions $$\{H_1, \ldots , H_n\}$${H1,…,Hn} on $$X$$X which are continuous on all of $$X$$X, smooth on an open dense subset $$U$$U of $$X$$X, and pairwise Poisson-commute on $$U$$U. Moreover, our integrable system in fact generates a Hamiltonian torus action on $$U$$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’ $$\mu = (H_1, \ldots , H_n): X \rightarrow \mathbb {R}^n$$μ=(H1,…,Hn):X→Rn is precisely the Newton-Okounkov body$$\Delta = \Delta (R, v)$$Δ=Δ(R,v) associated to the homogeneous coordinate ring $$R$$R of $$X$$X, and an appropriate choice of a valuation $$v$$v on $$R$$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 $$X$$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.