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

TL;DR

This paper establishes a geometric framework connecting polarizations, toric degenerations, and Newton-Okounkov bodies, demonstrating a continuous deformation of bases in geometric quantization that converges to delta functions at Bohr-Sommerfeld fibers.

Contribution

It introduces a new method to deform complex structures and bases in geometric quantization, linking toric degenerations with Newton-Okounkov bodies, and generalizes previous polarization independence results.

Findings

Constructed a deformation of complex structures and bases approaching delta functions.

Showed the hypotheses hold in general via Newton-Okounkov bodies.

Applied results to flag varieties and canonical bases, connecting to lattice points in polytopes.

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 $\Delta_{\underline{w}_0}(\lambda) \cap \mathbb{Z}^{\dim(G/B)}$.