Journal of Symplectic Geometry

Volume 12 (2014)

Number 3

Convergence of Kähler to real polarizations on flag manifolds via toric degenerations

Pages: 473 – 509

DOI: http://dx.doi.org/10.4310/JSG.2014.v12.n3.a3

Authors

Mark D. Hamilton (Department of Mathematics and Computer Science, Mount Allison University, Sackville, New Brunswick, Canada)

Hiroshi Konno (Graduate School of Mathematical Sciences, University of Tokyo, Japan; and Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki, Japan)

Abstract

In this paper, we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Our construction is based on a toric degeneration of flag varieties and a deformation of Kähler structure on toric varieties by symplectic potentials.

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