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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 3, Pages 15–50 (Mi izv334)  

This article is cited in 15 scientific papers (total in 16 papers)

Abelian Lagrangian algebraic geometry

A. L. Gorodentseva, A. N. Tyurinb

a Independent University of Moscow
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper begins a detailed exposition of a geometric approach to quantization, which is presented in a series of preprints ([23], [24], …) and which combines the methods of algebraic and Lagrangian geometry. Given a prequantization $U (1)$-bundle $L$ on a symplectic manifold $M$, we introduce an infinite-dimensional Kähler manifold $\mathscr P^{\mathrm{hw}}$ of half-weighted Planck cycles. With every Kähler polarization on $M$ we canonically associate a map $\mathscr P^{\mathrm{hw}}\overset{\gamma}{\to}H^{0}(M,L)$ to the space of holomorphic sections of the prequantization bundle. We show that this map has a constant Kähler angle and its “twisting” to a holomorphic map is the Borthwick–Paul–Uribe map. The simplest non-trivial illustration of all these constructions is provided by the theory of Legendrian knots in $S^3$.

DOI: https://doi.org/10.4213/im334

Full text: PDF file (3225 kB)
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English version:
Izvestiya: Mathematics, 2001, 65:3, 437–467

Bibliographic databases:

MSC: 53D50, 53C15
Received: 15.08.2000

Citation: A. L. Gorodentsev, A. N. Tyurin, “Abelian Lagrangian algebraic geometry”, Izv. RAN. Ser. Mat., 65:3 (2001), 15–50; Izv. Math., 65:3 (2001), 437–467

Citation in format AMSBIB
\Bibitem{GorTyu01}
\by A.~L.~Gorodentsev, A.~N.~Tyurin
\paper Abelian Lagrangian algebraic geometry
\jour Izv. RAN. Ser. Mat.
\yr 2001
\vol 65
\issue 3
\pages 15--50
\mathnet{http://mi.mathnet.ru/izv334}
\crossref{https://doi.org/10.4213/im334}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1853364}
\zmath{https://zbmath.org/?q=an:1005.53057}
\transl
\jour Izv. Math.
\yr 2001
\vol 65
\issue 3
\pages 437--467
\crossref{https://doi.org/10.1070/IM2001v065n03ABEH000334}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-27844480545}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. A. Tyurin, “Dynamical correspondence in algebraic Lagrangian geometry”, Izv. Math., 66:3 (2002), 611–629  mathnet  crossref  crossref  mathscinet  zmath
    2. F. A. Bogomolov, A. L. Gorodentsev, V. A. Iskovskikh, Yu. I. Manin, V. V. Nikulin, D. O. Orlov, A. N. Parshin, V. Ya. Pidstrigach, A. S. Tikhomirov, N. A. Tyurin, I. R. Shafarevich, “Andrei Nikolaevich Tyurin (obituary)”, Russian Math. Surveys, 58:3 (2003), 597–605  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. N. A. Tyurin, “Irreducibility of the ALG(a)-Quantization”, Proc. Steklov Inst. Math., 241 (2003), 249–255  mathnet  mathscinet  zmath
    4. N. A. Tyurin, “Letter to the editors”, Izv. Math., 68:3 (2004), 643–644  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. N. A. Tyurin, “Algebraic Lagrangian geometry: three geometric observations”, Izv. Math., 69:1 (2005), 177–190  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. N. A. Tyurin, “Existence theorem for the moduli space of Bohr–Sommerfeld Lagrangian cycles”, Russian Math. Surveys, 60:3 (2005), 572–574  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    7. Paoletti R., “Semiclassical almost isometry”, Lett. Math. Phys., 78:2 (2006), 189–204  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Debernardi M., Paoletti R., “Equivariant asymptotics for Bohr-Sommerfeld Lagrangian submanifolds”, Comm. Math. Phys., 267:1 (2006), 227–263  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. N. A. Tyurin, “Universal Maslov class of a Bohr–Sommerfeld Lagrangian embedding into a pseudo-Einstein manifold”, Theoret. and Math. Phys., 150:2 (2007), 278–287  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. S. A. Belev, N. A. Tyurin, “Nontoric Foliations by Lagrangian Tori of Toric Fano Varieties”, Math. Notes, 87:1 (2010), 43–51  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    11. Nikolay A Tyurin, “Exotic Chekanov tori in toric symplectic varieties”, J. Phys.: Conf. Ser, 411 (2013), 012028  crossref  isi  scopus
    12. N. A. Tyurin, “Special Bohr–Sommerfeld Lagrangian submanifolds”, Izv. Math., 80:6 (2016), 1257–1274  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. N. A. Tyurin, “Pseudotoric structures: Lagrangian submanifolds and Lagrangian fibrations”, Russian Math. Surveys, 72:3 (2017), 513–546  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. Barron T., Pollock T., “Kahler Quantization and Entanglement”, Rep. Math. Phys., 80:2 (2017), 217–231  crossref  mathscinet  zmath  isi  scopus
    15. N. A. Tyurin, “Special Bohr–Sommerfeld Lagrangian submanifolds of algebraic varieties”, Izv. Math., 82:3 (2018), 612–631  mathnet  crossref  crossref  adsnasa  isi  elib
    16. Alluhaibi N., Barron T., “On Vector-Valued Automorphic Forms on Bounded Symmetric Domains”, Ann. Glob. Anal. Geom., 55:3 (2019), 417–441  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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