Symplectic reduction and Lagrangian submanifolds of~$\operatorname{Gr}(1, n)$

N. A. Tyurin

Sbornik. Mathematics, Tome 215 (2024) no. 10, pp. 1426-1439

Résumé

New examples of Lagrangian submanifolds of the complex Grassmannian $\operatorname{Gr}(1, n)$ with the standard Kähler form are presented. The scheme of their construction is based on two facts: first, we put forward a natural correspondence between the Lagrangian submanifolds of a symplectic manifold obtained by symplectic reduction and the Lagrangian submanifolds of a large symplectic manifold carrying a Hamiltonian action of some group, to which this reduction is applied; second, we show that for some choice of generators of the action of $\mathrm T^k$ on $\operatorname{Gr}(1, n)$, $k=2, \dots, n-1$, and for suitable values of the moment map there exists an isomorphism $\operatorname{Gr}(1, n)/\!/\mathrm T^k \cong \operatorname{tot}(\mathbb{P}(\tau) \times \dots \times\mathbb{P}(\tau) \to \operatorname{Gr}(1, n-k))$, where the total space of the Cartesian product of $k$ copies of the projectivization of the tautological bundle $\tau \to \operatorname{Gr}(1, n-k)$ is on the right. Combining these two facts we obtain a lower bound for the number of topologically distinct smooth Lagrangian submanifolds in the original Grassmannian $operatorname{Gr}(1, n)$. Bibliography: 5 titles.

Détail

Keywords: algebraic variety, symplectic form, Lagrangian submanifold, Grassmannian.