Example of a Moduli Space of $D$-Exact Lagrangian Submanifolds: Spheres in the Flag Variety for $\mathbb C^3$

Nikolay A. Tyurin

Informatics and Automation, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 311-323

Résumé

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.