On toric hyperkähler manifolds with compact complex submanifolds (Q2518780)

Let \(K\) be a subtorus of \(T^N=U(1)^N\) and restrict to \(K\) the natural action of \(T^N\) on the quaternionic space \(\mathbb H^N\), endowed with the usual complex structures \(\mathbf{I}, \mathbf{J}, \mathbf{K}\) given respectively by the left multiplication by the purely imaginary quaternions \(i,j\) and \(k\). Denote by \(\mathfrak k\) the Lie algebra of \(K\), by \(\mathfrak k^*\) its dual and by \(\mathfrak k^*_{\mathbb C}=\mathfrak k\otimes\mathbb C\) its complexification. Set finally \(\mu:=(\mu_I,\mu_{\mathbb C}):\mathbb H^N\to\mathfrak{k}^*\times\mathfrak{k}^*_{\mathbb C}\) to denote the hyper-Kähler moment map for the action of \(K\) on \(\mathbb H^N\). Whenever \((\alpha,\beta)\in\mathfrak{k}^*\times\mathfrak{k}^*_{\mathbb C}\) is a regular value of \(\mu\) and if \(K\) acts freely on \(\mu^{-1}(\alpha,\beta)\), then one obtains a toric hyper-Kähler manifold (see also, for example, \textit{R. Bielawski} and \textit{A. Dancer} [Commun. Anal. Geom. 8, No.~4, 727--760 (2000; Zbl 0992.53034)]) \[ X(\alpha,\beta):=\mu^{-1}(\alpha,\beta)/K. \] Moreover, consider the holomorphic map \[ \Psi: (X(\alpha,\beta),\mathbf{I})\to\mu_{\mathbb C}^{-1}(\beta)//K_{\mathbb C} \] induced by the affine quotient map \(p:\mu_{\mathbb C}^{-1}(\beta)\to\text{Spm}\mathbb C[\mu_{\mathbb C}^{-1}(\beta)]^{K_{\mathbb C}}\), where \(K_{\mathbb C}\) is the complexification of \(K\). The paper under review consists of three parts. In the first one, the author gives some construction of compact complex submanifolds of \((X(\alpha,\beta),\mathbf{I})\) that are invariant under the \(T^N\)-action (not necessarily the projective line as in \textit{Y. Aoto} [Osaka J. Math. 41, No.~3, 583--603 (2004; Zbl 1076.53060)], see also \textit{H. Konno} [Int. J. Math. 14, No.~3, 289--311 (2003; Zbl 1054.53069)], but also of higher dimension). The techniques are essentially taken from classical toric geometry: namely polytopes, hyperplane arrangements, etc. In the second part the author studies some properties of the map \(\Psi\). In particular \(\Psi\) turns out to be a resolution of singularities, that is proper and surjective and biholomorphic on a dense open subset. Finally, the third part of the paper is devoted to discussing when complex structures (inherited by unitary imaginary quaternions) on \(X(\alpha,\beta)\) are equivalent.