Geometry of completely integrable systems and their degenerations

2014 Fiscal Year Final Research Report

• Project/Area Number: 23740055
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Kagawa University
• Principal Investigator: NOHARA Yuichi 香川大学, 教育学部, 准教授 (60447125)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: 完全可積分系 / フレアー理論

Outline of Final Research Achievements

We construct a completely integrable system on the Grassmannian of two-planes in an n-space associated with any trivalent tree with n leaves, and compute the potential function for its Lagrangian torus fiber by using its toric degeneration. The potential functions for different trees are related by a rational coordinate change, and the corresponding moment polytopes are related by its "tropicalization".
The Gelfand-Cetlin system has non-torus Lagrangian fibers on some of the boundary strata of the moment polytope. We compute Floer cohomologies of such non-torus Lagrangian fibers in the cases of the three-dimensional full flag manifold and the Grassmannian of two-planes in a four-space.

Free Research Field

幾何学