{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T02:14:12Z","timestamp":1768529652401,"version":"3.49.0"},"reference-count":57,"publisher":"World Scientific Pub Co Pte Lt","issue":"09","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Rev. Math. Phys."],"published-print":{"date-parts":[[2014,10]]},"abstract":"<jats:p> We review some recent results on the problem of the choice of polarization in geometric quantization. Specifically, we describe the general philosophy, developed by the author together with his collaborators, of treating real polarizations as limits of degenerating families of holomorphic polarizations. We first review briefly the general framework of geometric quantization, with a particular focus on the problem of the dependence of quantization on the choice of polarization. The problem of quantization in real polarizations is emphasized. We then describe the relation between quantization in real and K\u00e4hler polarizations in some families of symplectic manifolds, that can be explicitly quantized and that constitute an important class of examples: cotangent bundles of Lie groups, abelian varieties and toric varieties. Applications to theta functions and moduli spaces of vector bundles on curves are also reviewed. <\/jats:p>","DOI":"10.1142\/s0129055x1430009x","type":"journal-article","created":{"date-parts":[[2014,9,24]],"date-time":"2014-09-24T02:41:56Z","timestamp":1411526516000},"page":"1430009","source":"Crossref","is-referenced-by-count":3,"title":["Degenerating K\u00e4hler structures and geometric quantization"],"prefix":"10.1142","volume":"26","author":[{"given":"Jo\u00e3o P.","family":"Nunes","sequence":"first","affiliation":[{"name":"Center for Mathematical Analysis, Geometry and Dynamical Systems and Department of Mathematics, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal"}]}],"member":"219","published-online":{"date-parts":[[2014,10,15]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1142\/S0129167X98000282"},{"key":"rf2","unstructured":"M.\u00a0Abreu, Symplectic and Contact Topology: Interactions and Perspectives, Fields Institute Communications\u00a035, eds. Y.\u00a0Eliashberg, B.\u00a0Khesin and F.\u00a0Lalonde (Amer. Math. Soc., 2003)\u00a0pp. 1\u201324."},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.4310\/jdg\/1214446565"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-7.1.414"},{"key":"rf6","unstructured":"A.\u00a0Beauville, Current Topics in Complex Algebraic Geometry, MSRI Publications\u00a028 (Cambridge University Press, 1995)\u00a0pp. 17\u201333."},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02099772"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.4310\/jdg\/1335207374"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2010.01.023"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.4310\/PAMQ.2010.v6.n2.a4"},{"key":"rf11","volume-title":"Minicourse on Toric Varieties","author":"Cox D.","year":"2001"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1070\/RM1978v033n02ABEH002305"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.5802\/aif.1223"},{"key":"rf14","unstructured":"A. C.\u00a0da Silva, Symplectic Geometry of Integrable Hamiltonian Systems (Birkh\u00e4user, 2003)\u00a0pp. 85\u2013173."},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.24033\/bsmf.2100"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.4310\/jdg\/1214437664"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1016\/0550-3213(89)90436-7"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005874"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1515\/9781400882427"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2004.10.021"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2005.12.007"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1006\/jfan.2001.3903"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-1236(03)00108-3"},{"key":"rf24","first-page":"283","volume":"246","author":"Florentino C.","year":"2004","journal-title":"Proc. Steklov Inst. Math."},{"key":"rf25","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/397\/07461"},{"key":"rf26","doi-asserted-by":"publisher","DOI":"10.1007\/BF00669766"},{"key":"rf27","doi-asserted-by":"publisher","DOI":"10.4310\/jdg\/1214455538"},{"key":"rf28","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(83)90092-7"},{"key":"rf29","volume-title":"Lectures on Riemann Surfaces","author":"Gunning R. C.","year":"1966"},{"key":"rf30","doi-asserted-by":"publisher","DOI":"10.1006\/jfan.1994.1064"},{"key":"rf31","doi-asserted-by":"publisher","DOI":"10.1007\/s002200050059"},{"key":"rf32","doi-asserted-by":"publisher","DOI":"10.1007\/s002200200607"},{"key":"rf33","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2008.06.022"},{"key":"rf34","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-7116-5"},{"key":"rf35","first-page":"vi+60","volume":"207","author":"Hamilton M.","year":"2010","journal-title":"Mem. Amer. Math. Soc."},{"key":"rf36","doi-asserted-by":"publisher","DOI":"10.1007\/BF02161419"},{"key":"rf38","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61497-2","volume-title":"The Analysis of Partial Differential Operators I","author":"H\u00f6rmander L.","year":"2003"},{"key":"rf39","doi-asserted-by":"publisher","DOI":"10.5802\/aif.2517"},{"key":"rf40","doi-asserted-by":"publisher","DOI":"10.1007\/BF02096964"},{"key":"rf41","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-76079-2"},{"key":"rf42","doi-asserted-by":"crossref","unstructured":"A. A.\u00a0Kirillov, Encyclopaedia of Mathematical Sciences, Dynamical Systems\u00a04 (Springer-Verlag, 1990)\u00a0pp. 137\u2013172.","DOI":"10.1007\/978-3-662-06793-2_2"},{"key":"rf43","doi-asserted-by":"publisher","DOI":"10.4310\/JSG.2013.v11.n4.a4"},{"key":"rf44","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-006-0043-z"},{"key":"rf45","doi-asserted-by":"publisher","DOI":"10.1007\/BF01390167"},{"key":"rf46","doi-asserted-by":"publisher","DOI":"10.24033\/asens.1751"},{"key":"rf47","doi-asserted-by":"publisher","DOI":"10.1063\/1.531970"},{"key":"rf48","doi-asserted-by":"publisher","DOI":"10.2307\/1970710"},{"key":"rf49","volume-title":"Convex Bodies in Algebraic Geometry: An Introduction to Toric Varieties","author":"Oda T.","year":"1988"},{"key":"rf50","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1979-0530048-1"},{"key":"rf52","doi-asserted-by":"publisher","DOI":"10.4310\/JSG.2010.v8.n2.a1"},{"key":"rf53","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-6066-0"},{"key":"rf54","unstructured":"J.\u00a0\u015aniatycki, Differential Geometrical Methods in Mathematical Physics, Lect. Notes in Math\u00a0570 (Springer-Verlag, 1975)\u00a0pp. 46\u201366."},{"key":"rf55","doi-asserted-by":"publisher","DOI":"10.1006\/aima.1993.1011"},{"key":"rf56","series-title":"CRM Monographs","doi-asserted-by":"crossref","DOI":"10.1090\/crmm\/021","volume-title":"Quantization, Field Theory and Theta Functions","volume":"21","author":"Tyurin A.","year":"2003"},{"key":"rf57","series-title":"Panoramas et Synth\u00e8ses 2","volume-title":"Sym\u00e9trie Miroir","author":"Voisin C.","year":"1996"},{"key":"rf58","doi-asserted-by":"publisher","DOI":"10.1007\/BF02099122"},{"key":"rf59","doi-asserted-by":"publisher","DOI":"10.1007\/BF02099391"},{"key":"rf60","volume-title":"Geometric Quantization","author":"Woodhouse N. M. J.","year":"1991"}],"container-title":["Reviews in Mathematical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0129055X1430009X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T13:25:32Z","timestamp":1565097932000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0129055X1430009X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10]]},"references-count":57,"journal-issue":{"issue":"09","published-online":{"date-parts":[[2014,10,15]]},"published-print":{"date-parts":[[2014,10]]}},"alternative-id":["10.1142\/S0129055X1430009X"],"URL":"https:\/\/doi.org\/10.1142\/s0129055x1430009x","relation":{},"ISSN":["0129-055X","1793-6659"],"issn-type":[{"value":"0129-055X","type":"print"},{"value":"1793-6659","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,10]]}}}