Convergence of polarizations, toric degenerations, and Newton-Okounkov
bodies
Academic Article

Overview

Research

View All

Overview

abstract

Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$
and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$
satisfying some natural technical hypotheses, we construct a deformation
$\{J_s\}$ of the complex structure on $X$ and bases $\mathcal{B}_s$ of
$H^0(X,L, J_s)$ so that $J_0$ is the standard complex structure and, in the
limit as $s \to \infty$, the basis elements approach dirac-delta distributions
centered at Bohr-Sommerfeld fibers of a moment map associated to $X$ and its
toric degeneration. The theory of Newton-Okounkov bodies and its associated
toric degenerations shows that the technical hypotheses mentioned above hold in
some generality. Our results significantly generalize previous results in
geometric quantization which prove "independence of polarization" between
K\"ahler quantizations and real polarizations. As an example, in the case of
general flag varieties $X=G/B$ and for certain choices of $\lambda$, our result
geometrically constructs a continuous degeneration of the (dual) canonical
basis of $V_{\lambda}^*$ to a collection of dirac delta functions supported at
the Bohr-Sommerfeld fibres corresponding exactly to the lattice points of a
Littelmann-Berenstein-Zelevinsky string polytope
$\Delta_{\underline{w}_0}(\lambda) \cap \mathbb{Z}^{\dim(G/B)}$.