Upper bound for the Gromov width of coadjoint orbits of compact Lie groups (Q2827855)

Precisely, let \(T\) be the maximal torus of \(G\) and let \(\check{R}\) be the corresponding system of coroots. Let \(\mathfrak{g}\) and \(\mathfrak{t}\) be Lie algebras of \(G\) and \(T\), and let \(\mathcal{O}_\lambda\) be the coadjoint orbit passing through \(\lambda\in\mathfrak{t}^\ast\) and \(\omega_\lambda\) be the Kostant-Kirillov-Sourier form (explained in \S5) defined on \(\mathcal{O}_\lambda\), then the main theorem (\S1) of this paper is NEWLINE\[NEWLINE\mathrm{Gwidth}(\mathcal{O}_{\lambda,\omega_\lambda})\leq \min_{\check{\alpha}\in\check{R};(\lambda,\check{\alpha})\not=0}|(\lambda,\check{\alpha})|.NEWLINE\]NEWLINE Adopting the main theorem and Pabiniak's lower bound, the author remarks in the second theorem of \S1, that the Gromov width of a coadjoint orbit \(\mathcal{O}_\lambda\) of \(U(n)\) passing through \(\lambda=i\mathrm{diag}(\lambda_1,\ldots,\lambda_n)\in\mathfrak{u}(n)\cong\mathfrak{u}(n)^\ast\), is NEWLINE\[NEWLINE\mathrm{Gwidth}(\mathcal{O}_{\lambda,\omega_\lambda})=\min_{\lambda_i\not=\lambda_j}|\lambda_i-\lambda_j|.NEWLINE\]NEWLINE The proof of the main theorem is based on Gromov's estimate NEWLINE\[NEWLINE\mathrm{Gwidth}(M,\omega)\leq \omega(A),NEWLINE\]NEWLINE where \(A\in H_2(M;\mathbb{Z})\) is a certain nontrivial spherical class [\textit{M. Gromov}, Invent. Math. 82, 307--347 (1985; Zbl 0592.53025), Theorem 2.1]. The condition on \(A\) is described by Gromov-Witten invariants. This is proved in Theorem 4.5 after reviewing \(J\)-holomorphic tools which are used throughout the paper. In this paper, the definition of Gromov-Witten invariants follows that of \textit{K. Cieliebak} and \textit{K. Mohnke} [J. Symplectic Geom. 5, No. 3, 281--356 (2007; Zbl 1149.53052)].NEWLINENEWLINEAfter explaining the geometry of coadjoint orbits of compact Lie groups, including the definition of the Kostant-Kirilllov-Sourier form, the curve neighborhood is explained and it is indicated how to use it to compute Gromov-Witten invariants on homogeneous spaces in \S6. The author says that the materials presented here is mostly adopted from \textit{A. S. Buch} and \textit{L. C. Mihalcea} [J. Differ. Geom. 99, No. 2, 255--283 (2015; Zbl 1453.14117)]. The main theorem is proved for Grassmannian manifolds in \S7 (Theorem 7.2. Some technical part of the proof of Theorem 7.1, which is essential to the proof Theorem 7.2, is given in \S9, the last Section). This section is the technical core of the paper. The main theorem for arbitrary coadjoint orbits of compact Lie group is obtained by computing Gromov-Witten invariants on holomorphic fibrations where fibers are isomorphic to Grassmannian manifolds (\S8. Theorem 8.2).NEWLINENEWLINEThe author remarks, that the main theorem for \(regular\) coadjoint orbits of compact Lie groups was also found by \textit{M. Zoghi} [The Gromov width of coadjoint orbits of compact Lie groups. Toronto: University of Toronto (PhD Thesis) (2010)]. But in this paper, the main theorem is proved for arbitrary coadjoint orbits.