Poincaré duality and equivariant (co)homology. (Q5954582)

For a compact complex variety \(X\) with vanishing odd degree Betti numbers, the cohomology ring \(H^*(X,\mathbb{C})\) is a commutative, positively graded algebra of finite dimension as a complex vector space. Then, the dualizing module (in the sense of commutative algebra) of \(H^*(X,\mathbb{C})\) is the usual homology \(H_*(X,\mathbb{C})\), and \(H^*(X, \mathbb{C})\) is Gorenstein if \(X\) satisfies Poincaré duality.NEWLINENEWLINENEWLINE In the paper under review, the author investigates whether similar results hold in the more general setup of equivariant (co)homology. NEWLINEOn several interesting classes of varieties, like Schubert and flag varieties, there exists a torus action with isolated fixed points. Then, the equivariant cohomology ring \(H^*_T(X,\mathbb{C})\) is positively graded, commutative, reduced and Cohen--Macaulay. NEWLINEThe dualizing module of \(H_T^*(X,\mathbb{C})\) turns out to be the equivariant Borel--Moore homology \(H^T_*(X,\mathbb{C})\). NEWLINEBy studying the restriction map to the fixed point set \(H_T^*(X,\mathbb{C})\to H_T^*(X^T,\mathbb{C})\) and applying the localization theorem, the author obtains several linear inequalities on the Betti numbers of \(X\), generalizing a result of \textit{J.B. Carrell} [in: Algebraic groups and their generalizations: classical methods, Proc. Symp. Pure Appl. Math. 56, Part I, 53--61 (1994; Zbl 0818.14020)].