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We study harmonic bundles with an additional structure called symplectic structure. We study them for the case of the base manifold is compact and non-compact. For the compact case, we show that a harmonic bundle with a symplectic structure is equivalent to principle $Sp(2n,C)$-bundle with a reductive flat connection. For the non-compact case, we show that a polystable good filtered Higgs bundle with a perfect skew-symmetric pairing is equivalent to a good wild harmonic bundle with a symplectic structure., Comment: Comments are welcome! 23 pages

Let $X$ be a compact K\"ahler manifold and $(E,\overline\partial_E,\theta)$ be a Higgs bundle over it. We study the structure of the Kuranishi space for the pair $(X, E,\theta)$ when the Higgs bundle admits a harmonic metric or equivalently when the Higgs bundle is polystable and the Chern classes are 0. Under such assumptions, we show that the Kuranishi space of the pair $(X,E,\theta)$ is isomorphic to the direct product of the Kuranishi space of $(E,\theta)$ and the Kuranishi space of $X$. Moreover, when $X$ is a Riemann surface and $(E,\overline\partial_E,\theta)$ is stable and the degree is 0, we show that the deformation of the pair $(X,E,\theta)$ is unobstructed and calculate the dimension of the Kuranishi space., Comment: 15 pages. Comments are welcome! Typos are fixed

We study the deformation of the holomorphic-Higgs pair. The holomorphic-Higgs pair is a pair of a complex manifold and a Higgs bundle over it. We introduce the differential graded Lie algebra (DGLA) which comes from the deformation. We derive the Maurer-Cartan equation which governs the deformation of the holomorphic-Higgs pair, construct the Kuranishi family of it, and prove its local completeness., Comment: 25page. Comments are welcome!