Let \(X\) be an equidimensional variety (not necessarily irreducible) over an algebraically closed field (of any characteristic), and suppose that an algebraic group \(G\) acts on \(X\). In this article the authors study certain conditions for which the odd dimensional intersection cohomology of \(X\) vanishes. The idea is to use equivariant intersection cohomology. This article is a continuation of R. Joshua’s earlier paper with the same title [Math. Z. 195, 239-253 (1987; Zbl 0637.14014)], where condition is given for the odd dimensional middle intersection cohomology sheaves of \(X\) to vanish.
One of the main results of the present article is that if \(X\) has a finite number of orbits, and each orbit admits an attractive slice, then the stratification of orbits is perfect for equivariant intersection cohomology. Using this result, the authors prove several theorems relating the vanishing of global odd dimensional intersection cohomology and the vanishing of odd dimensional intersection cohomology sheaves with respect to equivariant local systems. For example, using the local structure for spherical varieties, it is shown that odd dimensional intersection cohomology groups and intersection cohomology sheaves vanish.