Torus actions and tensor products of intersection cohomology (Q2352721)

Let \(X\) be a smooth complex projective variety with an action of \(\mathbb{C}^*\). Let us assume that \textit{A. Bialynicki-Birula} decomposition of \(X\) [Ann. Math. (2) 98, 480-497 (1973; Zbl 0275.14007)] is a stratification. Denote by \(X_n\) the closures of strata and by \(L_{j,n}=(IC_{X_j})_{|X_n}\) the restrictions of the intersection sheaves. The map \[ H^*(X_n;L_{j_1,n})\otimes_{H^*(X)}\dots\otimes_{H^*(X)}H^*(X_n;L_{j_m,n}) \to H^*(X_n;L_{j_1,n}\otimes\dots\otimes L_{j_m,n}) \] induced by the cup product is studied. It is proved that under some additional geometric assumptions this map is an isomorphism. An analogous theorem is proved for equivariant cohomology. The proof is by induction on strata. The key argument allowing to split long exact sequences is the weight argument based on \textit{M. Saito}'s theory of mixed Hodge modules [Publ. Res. Inst. Math. Sci. 26, No. 2, 221--333 (1990; Zbl 0727.14004)].