This final chapter has a more informal character. On the one hand, we want to give some guidance for approaching recent publications on period domains (some of which of the present authors). On the other hand, we want to flesh out the remarks in the Introduction on some of the open problems in this area.

In the first section we explain the fundamental complex and the geometry around it. This is the crucial tool in the proof of Theorems 3.3.10, 7.2.4, 10.3.3 in the text, which are only stated in this book, but not proved. In the second section, we compare period domains over a finite field with Deligne-Lusztig varieties. The third section describes special features of the Drinfeld space for a local non-Archimedean field K, which are either not shared by other period domains, or are unknown in general. The fourth section describes the conjectural admissible analytic subset of period domains with its local system over it, and Hartl's candidate for it in a special case. In the fifth section we discuss the problem of determining the cohomology complex of period domains, and explain what is known in the Lubin–Tate and the Drinfeld cases.

The fundamental complex

In this section we give the construction of an acyclic resolution of an étale sheaf on the complement of a period domain. This complex appears in, and is the main ingredient for the determination of the individual cohomology groups of period domains (as opposed to the Euler–Poincaré characteristic).