Intersection theory on spherical varieties. (English) Zbl 0819.14019

Fulton, William; MacPherson, Robert; Sottile, F.; Sturmfels, Bernd

J. Algebr. Geom. 4, No. 1, 181-193 (1995).

Spherical varieties are varieties with reductive group actions containing dense orbits of Borel subgroups; there are many important examples including all flag manifolds and toric varieties. This paper gives a finite presentation for the Chow homology groups of any variety \(X\) on which a connected solvable linear algebraic group (for example a Borel subgroup of a reductive group) acts with only finitely many orbits. The authors show that the Künneth map \[ A_ * (X) \otimes A_ * (Y) \to A_ * (X \times Y) \] is then an isomorphism for any scheme \(Y\). In addition if \(X\) is complete they give a method to compute the operational Chow cohomology groups of \(X\) (the Kronecker duality map is shown to be an isomorphism).

Reviewer: F.Kirwan (Oxford)

MSC:

14L30	Group actions on varieties or schemes (quotients)
14C17	Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
20G10	Cohomology theory for linear algebraic groups

Keywords:

intersection theory; spherical varieties; reductive group actions; Chow homology groups