Complete symmetric varieties. II: Intersection theory. (English) Zbl 0596.14041
Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya/Jap. 1983, Adv. Stud. Pure Math. 6, 481-513 (1985).
[For the entire collection see Zbl 0561.00006.]
This paper is a sequel to part I [in: Invariant theory, Proc. 1st 1982 Sess., C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1–44 (1983; Zbl 0581.14041)], where the authors constructed a “minimal” compactification \(X\) of a symmetric homogeneous space \(G/H\), with \(X\) “wonderful” in the sense of Luna and Vust. The authors now construct a so-called space of conditions, \(C^*(G/H)\), which becomes a ring under a certain intersection product. This ring can be realized as the projective limit of the Chow rings (or, what is the same, the homology rings) of all wonderful compactifications \(X'\) dominating \(X\). The authors’ main transversality theorem for this “universal” Chow ring is the following: Given a cycle \(Y\subset G/H\) one can find \(X'\) as above such that the closure of \(Y\) in \(X'\) has proper intersection with the cycle \(Z'\) (of \(X'\)) sum of all closures of the \(G\)-orbits in \(X'\). The proof involves some results on torus embeddings.
This paper is a sequel to part I [in: Invariant theory, Proc. 1st 1982 Sess., C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1–44 (1983; Zbl 0581.14041)], where the authors constructed a “minimal” compactification \(X\) of a symmetric homogeneous space \(G/H\), with \(X\) “wonderful” in the sense of Luna and Vust. The authors now construct a so-called space of conditions, \(C^*(G/H)\), which becomes a ring under a certain intersection product. This ring can be realized as the projective limit of the Chow rings (or, what is the same, the homology rings) of all wonderful compactifications \(X'\) dominating \(X\). The authors’ main transversality theorem for this “universal” Chow ring is the following: Given a cycle \(Y\subset G/H\) one can find \(X'\) as above such that the closure of \(Y\) in \(X'\) has proper intersection with the cycle \(Z'\) (of \(X'\)) sum of all closures of the \(G\)-orbits in \(X'\). The proof involves some results on torus embeddings.
Reviewer: R.Piene
MSC:
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
| 14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |
| 14M17 | Homogeneous spaces and generalizations |
