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- W2950836984 abstract "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." @default.
- W2950836984 created "2019-06-27" @default.
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- W2950836984 date "2012-05-23" @default.
- W2950836984 modified "2023-09-27" @default.
- W2950836984 title "Integrable systems, toric degenerations and Okounkov bodies" @default.
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- W2950836984 doi "https://doi.org/10.48550/arxiv.1205.5249" @default.
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