We give a curvature formula of the L^2 metric on the direct image of the relative canonical bundle twisted by a holomorphic line bundle endowed with a positive singular metric whose inverse has Poincaré type singularities along a relative snc divisor. The result applies to families of log canonically polarized pairs. Moreover, we show that it improves the general positivity result of Berndtsson-Paun in a special situation of a big line bundle.

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