Examples of Mironov cycles in Grassmannians (Q2661752)

The article under review is concerned with the construction of new Lagrangian submanifolds in algebraic manifolds, in particular, complex Grassmannians \(\mathrm{Gr}_{\mathbb C}(k, n+1)\) that consist of all \(k\)-dimensional subspaces of \({\mathbb C}^{n+1}\). This construction is a generalization of Mironov's construction in [\textit{A. E. Mironov}, Sb. Math. 195, No. 1, 85--96 (2004; Zbl 1078.53079); translation from Mat. Sb. 195, No. 1, 89--102 (2004)]. As the main result, this article produces a family of smooth submanifolds that are isomorphic to the total space of the bundle \((\underline{S}^1 \times S^{k-1}(\tau^*))/\mathbb{Z}_2 \to \mathrm{Gr}_{\mathbb R}(k, n)\). Here, \(\underline{S}^1 \to\mathrm{Gr}_{\mathbb R}(k, n)\) denotes the trivial bundle and \(S^{k-1}(\tau^*) \to\mathrm{Gr}_{\mathbb R}(k, n)\) is a bundle with fiber isomorphic to the spherization of the fiber of the dual tautological bundle \(\tau^*\). Moreover, \(\mathbb Z_2\) represents the diagonal action on the direct sum of these two bundles. Considering particular values of \(k\), concrete examples are produced. For instance, when \(k =1\), one obtains a generalized Klein bottle investigated in [loc. cit.; \textit{P. E. Pushkar}, Funct. Anal. Appl. 34, No. 4, 288--292 (2000; Zbl 1072.53539); translation from Funkts. Anal. Prilozh. 34, No. 4, 64--70 (2000)]. For another instance, when \(k=2\), topologically the resulting Lagrangian submanifolds are the total space of a non-trivial \(\mathbb T^2\)-bundle over \(\mathrm{Gr}_{\mathbb R}(2,n)\): the standard Liouville torus, i.e., the \(\mathbb T^2\)-bundle over a point when \(n =2\); a \(\mathbb T^2\)-bundle over \(\mathbb{RP}^2\) when \(n = 3\); isomorphic to \(S^1 \times \mathbb{P}(T^*S^{n-1})\) for a general \(n\). The construction in this article starts from viewing \(\mathrm{Gr}_{\mathbb C}(k, n+1)\) inside the projective space \(\mathbb{P}(\wedge^k\mathbb{C}^{n+1})\) via the classical Plücker embedding and extracting moment maps \(\{F_1, \dots, F_m\}\) on \(\mathrm{Gr}_{\mathbb C}(k, n+1)\) (from the projective space), corresponding to an incomplete \(\mathbb T^m\)-action for a sufficiently small \(m\). In fact, this article considers the special case where \(m =1\). Then, one considers the intersection \(\mathrm{Gr}_{\mathbb R}(n, k+1) \cap \{F_1 = c\}\) for each \(c \in (0,1)\), where \(c\) is away from the only critical values 0 and 1 of \(F_1\). The desired Lagrangian submanifold is then given by the image of the \(\mathbb T^1\)-action on this intersection. Note that this construction has the potential to generate more (new) examples of Lagrangian submanifolds by using multiple moment maps or their linear combinations.