Integrable systems, toric degenerations and Okounkov bodies
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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.