Integrable systems, toric degenerations and Okounkov bodies

• 成果类型: Article
• 署名作者: Harada, Megumi; Kaveh, Kiumars
• 署名单位: McMaster University; Pennsylvania Commonwealth System of Higher Education (PCSHE); University of Pittsburgh
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0574-4
• 发表日期: 2015
• 页码: 927-985
• 关键词: multiplicities; VARIETIES; bases

摘要

Let X be a smooth projective variety of dimension n over C 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 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,H- . . ., H-n} on X which are continuous on all of X, smooth on an open dense subset of 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,H- . . ., H-n) : X -> 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 (Kahler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Aojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties , 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.