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

DOI10.4310/CAG.2021.V29.N5.A6zbMATH Open1490.53104arXiv1612.08981MaRDI QIDQ2063731

Publication date: 3 January 2022

Published in: Communications in Analysis and Geometry (Search for Journal in Brave)

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

m a t h c a l B_{s}

of

H^{0} (X, L, J_{s})

so that

J_{0}

is the standard complex structure and, in the limit as

s o i n f t y

, 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

l a m b d a

, our result geometrically constructs a continuous degeneration of the (dual) canonical basis of

V_{l a m b d a}^{*}

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

D e l t a_{u n d e r l i n e w_{0}} (l a m b d a) c a p m a t h b b Z^{d i m (G / B)}

.

Full work available at URL: https://arxiv.org/abs/1612.08981

zbMATH Keywords

degeneration geometric quantization Okounkov bodies

Mathematics Subject Classification ID

Toric varieties, Newton polyhedra, Okounkov bodies (14M25) Fibrations, degenerations in algebraic geometry (14D06) Divisors, linear systems, invertible sheaves (14C20) Geometric quantization (53D50)