Mathematische Annalen

, Volume 364, Issue 1–2, pp 1–28 | Cite as

Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds

  • William D. KirwinEmail author
  • José M. Mourão
  • João P. Nunes


Let \(P\) be a Delzant polytope. We show that the quantization of the corresponding toric manifold \(X_{P}\) in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time \(t = {\sqrt{-1}}s\). We relate the quantization of \(X_{P}\) in two different toric Kähler polarizations by taking the time-\({\sqrt{-1}}s\) Hamiltonian “flow” of strongly convex functions on the moment polytope \(P\). By taking \(s\) to infinity, we obtain the quantization of \(X_{P}\) in the (singular) real toric polarization. Recall that \(X_{P}\) has an open dense subset which is biholomorphic to \(({\mathbb {C}}^{*})^{n}\). The quantization of \(X_{P}\) in a toric Kähler polarization can also be described by applying the complexified Hamiltonian flow of the Abreu–Guillemin symplectic potential \(g\), at time \(t={\sqrt{-1}}\), to an appropriate finite-dimensional subspace of quantum states in the quantization of \(T^{*}{\mathbb {T}}^{n}\) in the vertical polarization. By taking other imaginary times, \(t= k {\sqrt{-1}}, k\in {\mathbb {R}}\), we describe toric Kähler metrics with cone singularities along the toric divisors in \(X_{P}\). For convex Hamiltonian functions and sufficiently negative imaginary part of the complex time, we obtain degenerate Kähler structures which are negative definite in some regions of \(X_{P}\). We show that the pointwise and \(L^2\)-norms of quantum states are asymptotically vanishing on negative-definite regions.



We would like to thank G. Marinescu for useful discussions and for showing us the proof of a semiclassical version of Theorem 5.10. We would also like to thank the referee for a very careful reading of the first draft of this article. J. M. Mourão and J. P. Nunes are supported by FCT/Portugal through the projects PEst-OE/EEI/LA009/2013, EXCL/MAT-GEO/0222/2012, PTDC/MAT/119689/2010, PTDC/MAT/1177762/2010. J. M. Mourão is thankful for generous support from the Emerging Field Project on Quantum Geometry from Erlangen–Nürnberg University.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • William D. Kirwin
    • 1
    Email author
  • José M. Mourão
    • 2
    • 3
  • João P. Nunes
    • 2
  1. 1.Mathematics InstituteUniversity of CologneCologneGermany
  2. 2.Department of Mathematics, Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  3. 3.Lehrstuhl für Theoretische Physik IIIFriedrichs-Alexander-UniversitätErlangen-NürnbergGermany

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