Compactification of varieties (Q2644723)

This paper is motivated by the work of \textit{G. W. Brumfiel} [Geometry of group representations, Proc. AMS-IMS-SIAM joint Summer Res. Conf., Boulder/Colo. 1987, Contemp. Math. 74, 51-75 (1988; Zbl 0662.32022)] on the compactification of the Teichmüller space, who was himself motivated by the work of Morgan and Shalen on this subject [\textit{J. W. Morgan} and \textit{P. B.Shalen}, Ann. Math., II. Ser. 120, 401-476 (1984; Zbl 0583.57005)]. An important feature of the compactification of complex or real varieties constructed by Morgan and Shalen is that the ideal points may be described in terms of Krull valuations (these valuations are used afterwards to construct trees). N. Schwartz introduces here a ``valuation spectrum'' associated to the variety. The valuations involved comprise Krull valuations and absolute values on the residue fields at prime ideals of the coordinate ring. The points of the variety are identified to closed points in this spectrum (taking the absolute value at this point). The compactification is built in this spectrum, and the ideal points are then immediately associated to Krull valuations. In the real case, the author's construction is strongly related to Brumfiel's, who uses the real spectrum. In the complex case, the construction uses a ``complex spectrum'' sitting inside the valuation spectrum. In both cases, the compactification of Morgan and Shalen factors through the author's compactification. (There are some misprints, for instance the ommission of absolute values in the definition of \(\alpha_ f\) on p. 353).