Small deformations of certain compact manifolds of class L (Q1078377)

As defined by M. Kato, a 3-dimensional complex manifold is of Class L if it contains a domain which is biholomorphic to a domain \(U_ r\) in \({\mathbb{P}}^ 3\) of the form \(U_ r=\{\zeta \in {\mathbb{P}}^ 3| | \zeta_ 0|^ 2+| \zeta_ 1|^ 2<r(| \zeta_ 2|^ 2+| \zeta_ 3|^ 2\quad)\}.\) If we set \(W={\mathbb{P}}^ 3-\ell_ 0-\ell_{\infty}\) with \(\ell_ 0=\{\zeta_ 0=\zeta_ 1=0\}\), \(\ell_{\infty}=\{\zeta_ 2=\zeta_ 3=0\}\) and if \(\alpha\) is a complex number with \(0<| \alpha | <1\), then \(g: (\zeta_ 0,\zeta_ 1,\zeta_ 2,\zeta_ 3)\to (\zeta_ 0,\zeta_ 1,\alpha \zeta_ 2,\alpha \zeta_ 3)\) generates an infinite cyclic group \(<g>\) acting on W, and \(M=W/<g>\) is a typical compact manifold of Class L. A remarkable property is that any two manifolds of Class L can be ''summed'' along neighborhoods of lines. Thus starting from the above M one can construct \(M(2)=M\#M\), \(M(3)=M(2)\#M\) etc. [\textit{M. Kato}, Jap. J. Math., New. Ser. 11, 1-58 (1985; Zbl 0588.32032)]. In the paper under review, the author studies small deformations of M(n) by Kodaira-Spencer's theory. First he calculates \(\dim H^ 1(M(n),\theta),\) where \(\theta\) denotes the tangent bundle, and then he constructs an explicit family which have correct number of independent parameters. In particular, any small deformation \(M_ t\) of \(M=M(1)\) is given in the form \(W/<g_ t>\) with \(g_ t\) being certain perturbation of g with 7 parameters.