50 research outputs found
Toric degenerations of Gelfand-Cetlin systems and potential functions
We define a toric degeneration of an integrable system on a projective
manifold, and prove the existence of a toric degeneration of the Gelfand-Cetlin
system on the flag manifold of type A. As an application, we calculate the
potential function for a Lagrangian torus fiber of the Gelfand-Cetlin system.Comment: 54 pages, 8 figures. v2: added section 4, revised section 9, and
minor changes here and ther
Deformation of singular curves on surfaces
In this paper, we consider deformations of singular complex curves on complex
surfaces. Despite the fundamental nature of the problem, little seems to be
known for curves on general surfaces. Let be a complete integral
curve on a smooth surface. Let be a partial normalization of ,
and be the induced map. In this paper, we
consider deformations of . The problem of the existence of
deformations will be reduced to solving a certain explicit system of polynomial
equations. This system is universal in the sense that it is determined solely
by simple local data of the singularity of , and does not depend on the
global geometry of or . Under a relatively mild assumption on the
properties of these equations, we will show that the map has
virtually optimal deformation property