Noncommutative Kähler structure on [formula omitted]-dynamical systems

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Notions of noncommutative complex and Kähler structure have been introduced by Fröhlich et al. (1999), in the context of supersymmetric quantum theory. Here we show that whenever a C∗-dynamical system (A,G,α,τ) equipped with a faithful G-invariant trace τ, where G is an even dimensional abelian Lie group, determines a spectral triple, the smooth dense subalgebra A∞ inherits a noncommutative Kähler structure. In particular, whenever T2n acts ergodically on the algebra, it inherits a noncommutative Kähler structure. This produces a class of examples of noncommutative Kähler manifolds. As a corollary, we obtain that all the noncommutative even dimensional tori are noncommutative Kähler manifolds. We explicitly compute the space of complex differential forms and study holomorphic vector bundles on all noncommutative even dimensional tori.