Convergence of polarizations, toric degenerations, and Newton–Okounkov bodies

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 $\lbrace J_s \rbrace$ 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 Kähler quantizations and real polarizations. As an example, in the case of general flag varieties $X = G/B$ and for certain choices of highest weight $\lambda$, our result geometrically constructs a continuous degeneration of the (dual) canonical basis of $V^{\ast}_\lambda$ 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 $\Delta_{\underline{w}_0}(\lambda) \cap \mathbb{Z}^{\dim(G/B)}$.

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The first author is partially supported by an NSERC Discovery Grant.

The second author is partially supported by an NSERC Discovery Grant and a Canada Research Chair (Tier 2) Award.

The third author is partially supported by a National Science Foundation Grant (Grant ID: DMS-1601303), Simons Foundation Collaboration Grant for Mathematicians, and Simons Fellowship award.

Received 26 February 2018

Accepted 19 January 2019

Published 1 December 2021