Geometry and topology of 4-manifolds

近十年来，人们对具有Lefschetz纤维化结构的4-流形进行了大量的研究。Lefschetz束的概念在20世纪20年代被引入代数几何中，但最近这个概念已经成为研究的中心主题，主要是由于唐纳森和Gompf的结果，根据Lefschetz纤维化是辛结构的基础结构。Lefschetz纤维化的结构是由它们在映射类群中取值的单值表示来拓扑描述的，而如上所述，它的结构与代数几何和辛几何密切相关。因此，Lefschetz纤维化的研究被认为是四维流形、曲面的映射类群、代数几何、辛几何等诸多研究领域的交汇点，本文正是考虑到这一点，注重不同研究领域之间的交流。例如，在研究期间，我们每年夏天都举办一次题为“拓扑和代数几何”的研究研讨会。我们也是本着同样的思想，举办了两次“拓扑与辛几何”专题讨论会，在Lefschetz纤维化、低维流形的重叠结构、单值性、图的组合学、映射类群（Teichmuller模群）、Riemann曲面的退化与分裂族的构造、拓扑与解析局部签名、Dedekind和、辛几何等方面取得了许多新的成果。通过这些结果，我们获得了一些新的研究前景。

Since a decade ago, there have been many research activities on 4-manifolds having the structure of Lefschetz fibrations. The concept of Lefschetz pencils was introduced in algebraic geometry in 1920's, but recently the concept has become the central theme of investigations mainly due to the results of Donaldson and Gompf according to which Lefschetz fibrations are the underlying structure of the symplectic structure. The structure of Lefschetz fibrations is topologically described by their monodromy representations taking the value in the mapping class groups, while the structure is, as mentioned above, closely related to algebraic geometry and symplectic geometry. Thus the study of Lefschetz fibrations is considered to be a meeting place of many research areas, such as 4-manifolds, mapping class groups of surfaces, algebraic geometry, symplectic geometry ad so on. In the present research, we took the above mentioned situation in consideration, and put stress on communication between different research areas. For example, we held a research seminar every summer during the research period entitled "Topology and Algebraic Geometry". Also by the same idea, we held two seminars entitled "Topology and Symplectic Geometry".We have many new results on Lefschetz fibrations, fibering structure of low dimensional manifolds, monodromy, combinatorics of charts, mapping class groups (Teichmuller modular groups), degenerations of Riemann surfaces and the construction of splitting families, topological and analytic local signature, Dedekind sums, and symplectic geometry. By these results, we have obtained some new prospects of research.