In this chapter we introduce period domains in their simplest version. We start with a vector space V of dimension n over a field k, and consider the variety ℱ = ℱ(V, v) of all ℝ-filtrations ℱ of a given type v. In the first section, we show that those filtrations ℱ of type v which violate the semi-stability condition along a given k-subspace V′ form a Zariski-closed subset. It then follows that, if k is a finite field, the semi-stable filtrations ℱ of type v form a Zariskiopen subset of ℱ. This open subset is the so-called period domain. It can be considered as a moduli space for semi-stable filtrations of type v on V.

The analogy with vector bundles on a curve is again a useful guide. Indeed, semi-stability of vector bundles was historically introduced to define a good moduli space via Geometric Invariant Theory. The first instance of this analogy will be a characterization of the period domain in terms of the Hilbert–Mumford criterion from GIT, in Section 2. The second instance will be the description of the natural stratification of the whole flag variety ℱ according to the Harder–Narasimhan type, similar to that of the moduli stack of vector bundles according to the HN type (see also the “Notes and References” of Section 3). In particular, we will clarify the structure of each HN-stratum in terms of period domains attached to vector spaces of smaller dimension.