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

Hamilton, Mark; Konno, Hiroshi

doi:10.4310/jsg.2014.v12.n3.a3

Cited by 10 publications

(31 citation statements)

References 7 publications

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“…Secondly, we intend to apply our results to provide explicit geometric interpretations of crystal bases for representations of connected reductive algebraic groups through the theory of geometric quantization and Bohr-Sommerfeld fibers (see e.g. [Bu10]) using methods similar to the work of Hamilton and Konno in [HaKo11]. Thirdly, it has been suggested to us by Milena Pabiniak and Yael Karshon that Theorem B can be applied to prove Biran's conjecture on the Gromov width of projective varieties (cf.…”

Section: We Note Thatmentioning

confidence: 99%

Integrable systems, toric degenerations and Okounkov bodies

2015

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Abstract. Let X be a smooth projective variety of dimension n over C equipped with a very ample Hermitian line bundle L. In the first part of the paper, we show that if there exists a toric degeneration of X satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from X to the special fiber X 0 which is a symplectomorphism on an open dense subset U . From this we are then able to construct a completely integrable system on X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions {H 1 , . . . , Hn} on X which are continuous on all of X, smooth on an open dense subset U of X, and pairwise Poisson-commute on U . Moreover, our integrable system in fact generates a Hamiltonian torus action on U . In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the 'moment map' µ = (H 1 , . . . , Hn) : X → R n is precisely the Newton-Okounkov body ∆ = ∆(R, v) associated to the homogeneous coordinate ring R of X, and an appropriate choice of a valuation v on R. Our main technical tools come from algebraic geometry, differential (Kähler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Lojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties X, this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.

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Section: We Note Thatmentioning

confidence: 99%

Integrable systems, toric degenerations and Okounkov bodies

2015

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“…Namely, they have given a one-parameter family of complex structures {J t } t∈[0,∞) and a basis {s m t } m∈∆∩t * Z of the space of holomorphic sections associated with the complex structure J t for each t such that each section s m t converges to the delta function section supported on the corresponding Bohr-Sommerfeld fiber as t goes to ∞. The similar result has been also obtained for flag manifolds in [19] and smooth irreducible complex algebraic varieties by [18]. But in [19] and [18] the convergence has been shown only for the non-singular Bohr-Sommerfeld fibers whereas in [3] it has been shown for all Bohr-Sommerfeld fibers.…”

Section: Motivation and Main Theoremsmentioning

confidence: 57%

Adiabatic limits, Theta functions, and geometric quantization

Yoshida¹

2019

Preprint

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Let π : (M, ω) → B be a (non-singular) Lagrangian torus fibration on a compact, complete base B with prequantum line bundle (L, ∇ L ) → (M, ω). For a positive integer N and a compatible almost complex structure J on (M, ω) invariant along the fiber of π, let D be the associated Spin c Dirac operator with coefficients in L ⊗N . Then, we give an orthogonal family { ϑ b } b∈B BS of sections of L ⊗N indexed by the Bohr-Sommerfeld points B BS , and show that each ϑ b converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber π −1 (b) and the L 2 -norm of D ϑ b converges to 0 by the adiabatic(-type) limit. Moreover, if J is integrable, we also give an orthogonal basis {ϑ b } b∈B BS of the space of holomorphic sections of L ⊗N indexed by B BS , and show that each ϑ b converges to a deltafunction section supported on the corresponding Bohr-Sommerfeld fiber π −1 (b) by the adiabatic(-type) limit. We also explain the relation of ϑ b with Jacobi's theta functions when (M, ω) is T 2n . Contents Motivation and MainTheorems 1 1.1. Notations 4 2. Developing Lagrangian fibrations 4 2.1. Integral affine structures 4 2.2. Lagrangian fibrations 7 2.3. Lagrangian fibrations with complete bases 9 2.4. The lifting problem of the Γ-action to the prequantum line bundle 12 3. Degree-zero harmonic spinors and integrability of almost complex structures 15 3.1. Bohr-Sommerfeld points 15 3.2. Almost complex structures 16 3.3. The existence condition of non-trivial harmonic spinors of degree-zero 18 3.4. The Γ-equivariant case 27 4. The integrable case 29 4.1. Definition and properties of ϑ m N 29 4.2. The case when Z is constant 32 4.3. Adiabatic-type limit 33 5. The non-integrable case 36 References 40

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“…In ongoing work, we plan to use our description of the topology of the singular GZ fibers, and a description of a local model for the GZ systems in neighbourhoods of said fibers, to give a more principled justification of Guillemin and Sternberg's observation, i.e., to prove that a singular fiber is Bohr-Sommerfeld if and only if it is integral. Moreover, we hope to extend the results of Hamilton-Kono [HK14] by showing that holomorphic sections in the Kähler quantization that correspond to boundary points of the GZ polytope converge under a deformation of complex structure to distributional sections supported on the singular fibers over the same boundary points. Organization of the paper.…”

Section: Introductionmentioning

confidence: 83%

“…Gelfand-Zeitlin systems are a family of completely integrable systems named for their connection to Gelfand-Zeitlin canonical bases [GS83a], 1 most commonly studied on coadjoint orbits of unitary and orthogonal Lie groups. The fibers of their moment maps, or Gelfand-Zeitlin fibers, are interesting from several perspectives, such as geometric quantization [GS83a,HK14], Floer theory [NNU10, NU16, CKO20], and the topology of integrable systems on symplectic manifolds [BMMT18, Problem 2.9]. The moment map images of Gelfand-Zeitlin systems on unitary and orthogonal coadjoint orbits are polytopes known as Gelfand-Zeitlin polytopes whose faces are enumerated by combinatorial diagrams called Gelfand-Zeitlin patterns 2 (see Figure 4).…”

Section: Introductionmentioning

confidence: 99%

The topology of Gelfand-Zeitlin fibers

Carlson¹,

Lane²

2021

Preprint

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We prove several new results about the topology of fibers of Gelfand-Zeitlin systems on unitary and orthogonal coadjoint orbits. First, we provide a simplified description of their diffeomorphism types as balanced products, recovering results of Bouloc-Miranda-Zung as special cases. Second, we compute their cohomology rings. Finally, we complete the computation of their first three homotopy groups (the first and second homotopy groups were described by Cho-Kim-Oh in the unitary case). Our results build on Cho-Kim-Oh's description of Gelfand-Zeitlin fibers as total spaces of certain towers of fiber bundles. Our description of the diffeomorphism type, cohomology ring, and homotopy groups can be read in a straightforward manner from the combinatorics of the associated Gelfand-Zeitlin pattern.

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

Cited by 10 publications

References 7 publications

Integrable systems, toric degenerations and Okounkov bodies

Integrable systems, toric degenerations and Okounkov bodies

Adiabatic limits, Theta functions, and geometric quantization

The topology of Gelfand-Zeitlin fibers

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