Complex structures of toric hyper-Kähler manifolds (Q1763041)

The author studies complex structures on toric hype-Kähler manifolds [\textit{R.~Bielawski} and \textit{A.~Dancer}, Commun. Anal. Geom. 8, No. 4, 727--760 (2000; Zbl 0992.53034)]. A toric hyper-Kähler manifold is a special type of a hyper-Kähler quotient [\textit{N.~Hitchin}, \textit{A.~Karlhede}, \textit{U.~Lindström} and \textit{M.~Roček}, Commun. Math. Phys. 108, 535--589 (1987; Zbl 0612.53043)]. One considers a subtorus \(K\) of the real torus \(T^N\), which acts on the \(N\)-dimensional quaternionic space \(H^n\) from the right. One then defines the hyper-Kähler moment map \(\mu_K\). For each regular value \(\nu\) of \(\mu_K\) for which \(K\) acts freely on the fibre of \(\mu_K\) over \(\nu\), the quotient \(X(\nu)\) of the fibre by the action of \(K\) is a so-called toric hyper-Kähler manifold. As a hyper-Kähler manifold, \(X(\nu)\) comes with a family of complex structures parametrised by the two-dimensional sphere \(S^2\). The author proves a theorem claiming the biholomorphicity of \(X(\nu)\) with respect to one complex structure and \(X(\nu')\) with respect to another, when \(\nu\) and \(\nu'\) are related by an orthogonal transformation. He then determines for a given \(\nu\) such that \(X(\nu)\) is a toric hyper-Kähler manifold explicitly all those complex structures \(I\) in the \(S^2\)-family such that \((X(\nu), I)\) allows a compact complex submanifold, in fact a projective line. There is always an even, finite number of these complex structures. All the other complex structures lead to an affine complex manifold. The author then considers the case that \(K\) is one-dimensional and the case that \(K\) is one-codimensional as examples and applies his theorems to these cases. In the last chapter he specialises to the case that there are exactly two complex structures on \(X(\nu)\) such that each of the corresponding complex manifolds contains a compact complex submanifold. Then the author is able to prove that all other complex structures lead to the same affine complex variety. The question whether the complex structures leading to affine varieties are pairwise biholomorphic in case there are more than two complex structures allowing compact complex submanifolds is still open.