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A flat Higgs bundle structure on the complexified Kähler cone

We shall construct a natural Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of the flat-ness of our Higgs bundle, we find a commutator identity that can be used to decode the variational properties of the polarized Hodge-Lefschetz module structure on the fibres of our Higgs bundle. Thus we can use a generalized version of Lu's Hodge metric to study the curvature property of the complexified Kähler cone. In particular, it implies that the above Hodge metric defines a Kähler metric on the complexified Kähler cone with negative holomorphic sectional curvature, which can be seen as a new result on Wilson's conjecture.