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Kähler structure on certain $C^*$-dynamical systems and the noncommutative even dimensional tori

Let $G$ be an even dimensional, connected, abelian Lie group and $(\mathcal{A}^\infty,G,α,τ)$ be a $C^*$-dynamical system equipped with a faithful $G$-invariant trace $τ$. We show that whenever it determines a $\varTheta$-summable even spectral triple, $\mathcal{A}^\infty$ inherits a Kähler structure. Moreover, there are at least $\prod_{j=1,\,j\,odd}^{\,dim(G)}(dim(G)-j)$ different Kähler structures. In particular, whenever $\mathbb{T}^{2k}$ acts ergodically on the algebra, it inherits a Kähler strcture. This gives a class of examples of noncommutative Kähler manifolds. As a corollary, we obtain that all the noncommutative even dimensional tori, like their classical counterpart the complex tori, are noncommutative Kähler manifolds. We explicitly compute the space of complex differential forms for the noncommutative even dimensional tori and show that the category of holomorphic vector bundle over it is an abelian category. We also explain how the earlier set-up of Polishchuk-Schwarz for the holomorphic structure on noncommutative two-torus follows as a special case of our general framework.