Deformations of symplectic structures by moment maps (Q1784688)

A deformation theory of symplectic structures based on the quasi-Poisson Hamiltonian actions is presented. A moment map \(\mu\) for the action of a quasi-Poisson action satisfies a condition not for one quasi-Poisson structure but for a family of quasi-Poisson structures parametrized by elements \(t\in\Lambda^2\mathfrak{g}\). where \((\mathfrak{d},\mathfrak{g},\mathfrak{h})\) is the Manin triple, the infinitesimal version of the quasi-triple \((D,G,h)\), named twist (details are explained in \S2). By using the twist, the following theorem and definition of the deformation of symplectic structure are given (\S3): {Theorem 14}. Let \((M,\omega)\) be a symplectic manifold on which a compact Lie group \(G\) with Lie algebra \(\mathfrak{g}^\ast\) acts by a symplectic Hamiltonian action \(\sigma\), and \(\mu\) a moment map for \(\sigma\). If a twist \(t\) satisfies \([t.t]_M=0\), and the isotropic complement \(\mathfrak{g}_t^\ast\) is admissible on \(\mu(M)\), then \(t\) deforms \(\omega\) to a symplectic structure \(\omega^t\) induced by the non-degenerate Poisson structure \(\pi^t_M=\pi-t_M\). Then the following sufficient condition for a twist to deform a symplectic structure in the sense of this theorem is given: {Theorem 16}. Let \((M,\omega)\) \(G\) and \(\sigma\) are as in Theorem 14. Assume that \(X,Y\) in \(\mathfrak{g}\) satisfy \([X,Y]=0\). Then the twist \(t=\frac{1}{2}X\wedge Y\) in \(\Lambda^2\mathfrak{g}\) deforms the symplectic structure \(\omega\) to a symplectic structure \(\omega_t\). For example, a twist in \(\Lambda^2\mathfrak{h}\), where \(\mathfrak{h}\) is the Cartan subalgebra of \(\mathfrak{g}\), satisfies the assumption of the theorem. As examples, deformations of \(\mathbb{CP}^n\) with the Fubini-Study form and the action of \(\mathrm{SU}(2)\), and the complex Grassmannian with the Kirillov-Kostant form with the action of \(\operatorname{SU}(n+1)\) are studied in \S4. As for \(\mathbb{CP}^1\), taking \(t=\sum_{i<j}\frac{1}{2}\lambda_{ij}e_i\wedge e_j, \lambda_{12}^2+\lambda_{13}^2+\lambda_{23}^2<1\), the Fubini-Study form is deformed to a symplectic form \(\omega_{\mathrm{FS}}^{\lambda t}\) whose symplectic volume is \[ \mathrm{vol}(\mathbb{CP}^1,\omega_{\mathrm{FS}}^{\lambda t})=\frac{\pi}{\lambda}\log\left|\frac{2+\lambda}{2-\lambda}\right|, \;\lambda\not=0. \] In \S5, the last section, deformations of symplectic toric manifolds are discussed and proves: {Theorem 21}. For any \(2n\)-dimensional compact connected symplectic toric manifold \((M,\omega)\) such that \[ \lambda\omega^{n-1}:H^1_{DR}(M;\mathbb{R})\to H_{DR}^{2n-1}(M,\mathbb{R}) \] is an isomorphism, and for any twist \(t\in\Lambda^2\mathbb{R}^n\), the manifold \((M,\omega^t)\) deformed by \(t\) is isomorphic to \((M,\omega)\) as a symplectic toric manifold. Before explaining these results, the quasi-Poisson theory [\textit{A. Alekseev} and \textit{Y. Kosmann-Schwarzbach}, J. Differ. Geom. 56, No. 1, 133--165 (2000; Zbl 1046.53055)], which is the base of this paper, is reviewed in \S2. Especially, it is remarked that the Manin quasi-triple \((\mathfrak{d},\mathfrak{g}.\mathfrak{h})\) defines the decomposition \(\mathfrak{d}=\mathfrak{g}\oplus\mathfrak{h}\), but \(\mathfrak{h}\) may not be unique. Therefore there may exist maps \(j: \mathfrak{g}^\ast\to \mathfrak{h}\) and \(j^\prime:\mathfrak{g}^\ast\to \mathfrak{h}^\prime\). The twist \(t\) is defined by \(j^\prime-j:\mathfrak{g}^\ast\to\mathfrak{d}\).