Kodaira fibred surfaces are remarkable examples of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. Our tour through algebraic surfaces and their moduli (with results valid also for higher dimensional varieties) deals with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (there are by the way rigid Kodaira fibrations), projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance remarkable surfaces constructed from VHS, surfaces isogenous to a product with automorphisms acting trivially on cohomology, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces.

小平纤维表面是射影分类空间的显着例子，并且仍然存在许多有趣的悬而未决的问题，特别是斜率问题。 Kodaira 纤维的拓扑表征是拓扑方法在曲面模空间和高维复代数簇及其紧致化研究中的应用的象征。我们对代数曲面及其模的游览（结果也适用于更高维的变体）涉及纤维化、映射类组中的单性和因式分解问题、Hodge 结构变分的新旧结果，特别是最近给出的答案（在与 Dettweiler 合作）解决藤田提出的长期存在的问题。在我们的游览景观中，伽罗瓦覆盖、变形和刚性流形（顺便说一下，还有刚性小平纤维）、射影分类空间、绝对伽罗瓦群在模空间上的作用，也站在最前沿。这些问题引出了有趣的代数曲面，例如由 VHS 构造的显着曲面、与自同构对上同调作用微不足道的乘积同构的曲面、Bagnera-de Franchis 簇中的超曲面、Inoue 型曲面。