Constructing hyper-Kähler manifolds by the method of S. G. Gindikin (Q1916674)

The author has generalized a construction of \textit{S. G. Gindikin} [Funct. Anal. Appl. 20, 238-240 (1986); translation from Funkts. Anal. Prilozh. 20, No. 3, 82-83 (1986; Zbl 0641.53063)] of hyper-Kähler manifolds. The basic concept of the paper is a fibre bundle \(Z\to C\), where \(Z\) is a complex manifold, \(C\) is a Riemann surface and the fibres are endowed with the complex symplectic structure given up to a factor. In the category of such bundles where \(C\) is a fixed compact Riemann surface, the author constructs a morphism, called an elementary SG-process, by means of three operations: projectivization, cut-out, and glueing of complex manifolds. He introduces the concept of a section coherent with SG-process (meant as a composition of a number of elementary SG-processes) and by means of these concepts he defines a fundamental construction of a certain fibre bundle and concludes by a classification of symplectic vector bundles on \(\mathbb{C}\mathbb{P}^1\), their Lagrange subspaces and SG-processes.