Study of Lagrangian fibrations via projective embeddings

证明了a型旗形上的完全可积系统Gelfand-Cetlin系统可以变形为环形上的环矩映射（这是与T. Nishinou和K. Ueda共同完成的工作）。作为应用，我们计算了Gelfand-Cetlin系统拉格朗日环面纤维的势函数，并证明了它与旗形的Landau-Ginzburg反射镜的超势相吻合。已知复向量空间中两平面格拉斯曼算子的环向退化是用一定的树来分类的。对于每一个这样的树，我都在Grassmannian上构造了一个完全可积系统，并证明了在相应的环退化下，它可以变形成一个环矩映射。这一结果表明，由Kapovich和Millson构造的多边形空间上的完全可积系统也允许变形为环矩映射。我还研究了投影线上2阶抛物束模空间上kapoovich - milson的可积系统和Goldman的可积系统之间的关系。

I have proved that the Gelfand-Cetlin system, a completely integrable system on a flag manifold of type A, can be deformed into a toric moment map on a toric variety (this is a joint work with T. Nishinou and K. Ueda). As an application we computed the potential functions of the Lagrangian torus fibers of the Gelfand-Cetlin system, and showed that it coincides with the superpotential of the Landau-Ginzburg mirror of the flag manifold.It is known that toric degenerations of a Grassmannian of two-planes in a complex vector space are classified by certain trees. For each such tree I have constructed a completely integrable system on the Grassmannian, and proved that it can be deformed into a toric moment map under the corresponding toric degeneration. This result implies that completely integrable systems on a polygon space constructed by Kapovich and Millson also admit deformations into toric moment maps. I have also studied a relation between Kapovich-Milson's integrable systems and Goldman's integrable systems on a moduli space of parabolic bundles of rank 2 on a projective line.