Special Bohr–Sommerfeld geomery
- 作者: Tyurin N.A.1,2
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隶属关系:
- Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna
- Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
- 期: 卷 80, 编号 2 (2025)
- 页面: 123-164
- 栏目: Articles
- URL: https://journal-vniispk.ru/0042-1316/article/view/306748
- DOI: https://doi.org/10.4213/rm10219
- ID: 306748
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详细
This survey sums up a cycle of papers devoted to the construction of finite-dimensional moduli spaces points in which are certain special Lagrangian submanifolds of compact complex simply connected algebraic varieties. The starting point for this construction was the idea, due to A. Tyurin, to treat Largrangian submanifolds (or equivalence classes of such submanifolds) as mirror counterparts of stable vector bundles. Our constructions are based on the programme of abelian Lagrangian algebraic geometry developed by A. Tyurin and Gorodentsev 25 years ago. Since this programme was in its turn based on the Bohr–Sommerfeld Lagrangian geometry known in geometric quantization, we call our construction special Bohr–Sommerfeld geometry. The definitions arising in the course of work turn out to be closely connected with the theory of Weinstein domains, Eliashberg's conjectures, and many other concepts in symplectic geometry. The core conjecture that arose in our work and is confirmed by the available examples states that each moduli space of this type is in its turn an algebraic variety.Bibliography: 13 titles.
作者简介
Nikolai Tyurin
Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna; Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Email: ntyurin@theor.jinr.ru
Doctor of physico-mathematical sciences, Professor
参考
- А. Н. Тюрин, Геометрия векторных расслоений, Сборник избранных трудов, 1, Ин-т компьютерных исследований, М.–Ижевск, 2004, 356 с.
- А. Л. Городенцев, А. Н. Тюрин, “Абелева лагранжева алгебраическая геометрия”, Изв. РАН. Сер. матем., 65:3 (2001), 15–50
- D. A. Cox, Sh. Katz, Mirror symmetry and algebraic geometry, Math. Surveys Monogr., 68, Amer. Math. Soc., Providence, RI, 1999, xxii+469 pp.
- A. N. Tyurin, “Fano versus Calabi–Yau”, The Fano conference (Torino, 2002), Univ. Torino, Dipart. Mat., Torino, 2004, 701–734
- N. Hitchin, “Lectures on special Lagrangian submanifolds”, Winter school on mirror symmetry, vector bundles and Lagrangian submanifolds (Harvard Univ., Cambridge MA, 1999), AMS/IP Stud. Adv. Math., 23, Amer. Math. Soc., Providence, RI; Int. Press, Somerville, MA, 2001, 151–182
- Н. А. Тюрин, “Специальные бор–зоммерфельдовы лагранжевы подмногообразия”, Изв. РАН. Сер. матем., 80:6 (2016), 274–293
- Н. А. Тюрин, “Специальные бор–зоммерфельдовы лагранжевы подмногообразия в алгебраических многообразиях”, Изв. РАН. Сер. матем., 82:3 (2018), 170–191
- Н. А. Тюрин, “Специальная геометрия Бора–Зоммерфельда: вариации”, Изв. РАН. Сер. матем., 87:3 (2023), 184–205
- Н. А. Тюрин, “Многообразие модулей $D$-точных лагранжевых подмногообразий”, Сиб. матем. журн., 60:4 (2019), 907–921
- Н. А. Тюрин, “Пример многообразия модулей $D$-точных лагранжевых подмногообразий: сферы в многообразии флагов в $mathbb C^3$”, Труды МИАН, 320, Алгебра, арифметическая, алгебраическая и комплексная геометрия (2023), 311–323
- R. Harvey, H. B. Lawson, Jr., “Calibrated geometries”, Acta Math., 148 (1982), 47–157
- Ya. Eliashberg, “Weinstein manifolds revisited”, Modern geometry: a celebration of the work of Simon Donaldson, Proc. Sympos. Pure Math., 99, Amer. Math. Soc., Providence, RI, 2018, 59–82
- Ф. Гриффитс, Дж. Харрис, Принципы алгебраической геометрии, Мир, М., 1982, 864 с.
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