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We prove that one can obtain natural bundles of Lie algebras on rank twos-K\"ahler manifolds, whose fibres are isomorphic to so(s+1,s+1), su(s+1,s+1)and sl(2s + 2,\R). In the most rigid case (which includes complex tori andabelian varieties) these bundles have natural flat connections, whose flatglobal sections act naturally on cohomology. We also present several naturalexamples of manifolds which can be equipped with an s-K\"ahler structure withvarious levels of rigidity: complex tori and abelian varieties, cotangentbundles of smooth manifolds and moduli of pointed elliptic curves.

Natural Lie Algebra bundles on rank two s-K\"ahler manifolds, abelian varieties and moduli of curves