Moderate degenerations of algebraic surfaces.

*(English)*Zbl 0774.14032
Complex algebraic varieties, Proc. Conf., Bayreuth/Ger. 1990, Lect. Notes Math. 1507, 113-132 (1992).

[For the entire collection see Zbl 0745.00049.]

The autor defines permissible degenerations of complex surfaces over \(\Delta=\{t\in\mathbb{C}:| t|<1\}\) allowing in the central fibre (terminal) singularities slightly more general than those occurring in semistable degenerations. Permissible degenerations whose fibres have only isolated singularities are called moderate. The reason for introducing this concept is that for an arbitrary algebraic degeneration of nonruled surfaces \(f:X\to\Delta\), after a base change, there exists a bimeromorphically equivalent degeneration \(g:Y\to\Delta\), which is permissible, has smooth projective surfaces as fibres over \(\Delta-\{0\}\) and is minimal, in the sense that \(K_ YC\geq 0\) for any curve \(C\) contained in a fibre of \(g\). So permissible degenerations can be viewed as a 2-dimensional analogue of semistable degenerations of curves. The author studies the topology, the rational cohomology and its Hodge structures for moderate degenerations showing that they behave similarly to semistable ones. Moreover he classifies the central fibres of minimal moderate degenerations with general fibres of Kodaira dimension 0 or 1, showing that a new phenomenon can occur for the elliptic surfaces case, due to the confluences of a multiple fibre on non-multiple singular fibres.

The autor defines permissible degenerations of complex surfaces over \(\Delta=\{t\in\mathbb{C}:| t|<1\}\) allowing in the central fibre (terminal) singularities slightly more general than those occurring in semistable degenerations. Permissible degenerations whose fibres have only isolated singularities are called moderate. The reason for introducing this concept is that for an arbitrary algebraic degeneration of nonruled surfaces \(f:X\to\Delta\), after a base change, there exists a bimeromorphically equivalent degeneration \(g:Y\to\Delta\), which is permissible, has smooth projective surfaces as fibres over \(\Delta-\{0\}\) and is minimal, in the sense that \(K_ YC\geq 0\) for any curve \(C\) contained in a fibre of \(g\). So permissible degenerations can be viewed as a 2-dimensional analogue of semistable degenerations of curves. The author studies the topology, the rational cohomology and its Hodge structures for moderate degenerations showing that they behave similarly to semistable ones. Moreover he classifies the central fibres of minimal moderate degenerations with general fibres of Kodaira dimension 0 or 1, showing that a new phenomenon can occur for the elliptic surfaces case, due to the confluences of a multiple fibre on non-multiple singular fibres.

Reviewer: A.Lanteri (Milano)

##### MSC:

14J15 | Moduli, classification: analytic theory; relations with modular forms |

14D15 | Formal methods and deformations in algebraic geometry |

14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |

32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |

14E30 | Minimal model program (Mori theory, extremal rays) |

14J17 | Singularities of surfaces or higher-dimensional varieties |