On the equivariant formality of Kähler manifolds with torus group actions (Q2384748)

An algebraic model for a space is a commutative differential graded algebra (CDGA) that determines the rational homotopy type of the space. A space is called formal if its rational homotopy type is determined by its cohomology ring regarded as a CGDA with differential \(d=0\). A classical theorem of rational homotopy theory states that compact Kähler manifolds are formal [\textit{P. Deligne, P. Griffiths, J. Morgan} and \textit{D. Sullivan}, Invent. Math. 29, 245--274 (1975; Zbl 0312.55011)]. In this paper the author applies her previous work on equivariant rational homotopy theory to prove an equivariant version of this theorem for compact simply connected Kähler manifolds with a holomorphic action of a torus group \(\mathbb{T}\). The author starts by reviewing the algebraic model \(\underline{\varOmega}(X)\) introduced in [\textit{M. A. Mandell} and \textit{L. Scull}, Math. Z. 240, No.~2, 261--287 (2002; Zbl 1005.55003)], for a \(\mathbb{T}\)-space \(X\). Here \(\underline{\varOmega}(X)\) is a diagram of CGDA indexed on a category \(\mathcal{D}\) related to the orbit category of \(\mathbb{T}\). This model uses a version of the de Rham differential form functor applied to the Borel bundles of the fixed sets \(X^K\), \(K < \mathbb{T}\). Then, for a Kähler manifold \(M\), an alternate model \(\underline{\mathbf{A}}^{\bullet}_{\mathbb{T}}(M\)) is introduced using forms on \(M\) rather than forms on Borel spaces. Using this model, the proof of formality for compact Kähler manifolds in [\textit{P. Deligne, P. Griffiths, J. Morgan} and \textit{D. Sullivan}, op. cit.] is adapted to the equivariant setting. In the last section, two examples coming from linear actions of \(\mathbb{T}^1\) on complex projective spaces are presented in detail.