Geodesic
Example of a Moduli Space of $D$-Exact Lagrangian Submanifolds: Spheres in the Flag Variety for $\mathbb C^3$
Informatics and Automation, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 311-323.

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In previous papers we proposed a construction of the moduli space of $D$-exact Lagrangian submanifolds in algebraic varieties with respect to a very ample divisor. The points of the moduli space are Hamiltonian equivalence classes of Lagrangian submanifolds in the complements $X\setminus D$, where $D$ is a divisor from a complete linear system; by the very definition this moduli space is a covering of an open subset in the projective space $|D|$. We showed that these moduli spaces are smooth and Kähler, and we proposed a way to distinguish, in such a moduli space, certain stable components whose main supposed property is to be algebraic. In the present paper we find the stable component of the moduli space of Lagrangian spheres in the flag variety with an ample divisor equal to half the anticanonical bundle, and show that this component is an algebraic variety itself. 
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Nikolay A. Tyurin. Example of a Moduli Space of $D$-Exact Lagrangian Submanifolds: Spheres in the Flag Variety for $\mathbb C^3$. Informatics and Automation, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 311-323. http://geodesic.mathdoc.fr/item/TRSPY_2023_320_a13/

[1] Auroux D., “Asymptotically holomorphic families of symplectic submanifolds”, Geom. Funct. Anal, 7:6 (1997), 971–995 | DOI | MR | Zbl

[2] Cieliebak K., Eliashberg Ya., From Stein to Weinstein and back: Symplectic geometry of affine complex manifolds, Colloq. Publ., 59, Amer. Math. Soc., Providence, RI, 2012 | MR | Zbl

[3] Griffiths P., Harris J., Principles of algebraic geometry, J. Wiley Sons, New York, 1978 | MR | Zbl

[4] Nohara Y., Ueda K., Floer cohomologies of non-torus fibers of the Gelfand–Cetlin system, E-print, 2014, arXiv: 1409.4049 | MR

[5] N. A. Tyurin, “Special Lagrangian fibrations on the flag variety $F^3$”, Theor. Math. Phys., 167:2 (2011), 567–576 | DOI | DOI | MR | Zbl

[6] N. A. Tyurin, “On Lagrangian spheres in the flag variety $F^3$”, Math. Notes, 98:1–2 (2015), 348–351 | DOI | DOI | MR | Zbl

[7] N. A. Tyurin, “The moduli space of $D$-exact Lagrangian submanifolds”, Sib. Math. J., 60:4 (2019), 709–719 | DOI | MR | Zbl