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arXiv · 2005.01500

Positivity of direct images with a Poincar\'e type twist

Philipp Naumann
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Abstract

We consider a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a line bundle $\mathcal{L}\to \mathcal{X}$. Given that $\mathcal{L}^{-1}$ carries a singular hermitian metric that has Poincar\'e type singularities along a relative snc divisor $\mathcal{D}$, the direct image $f_*(K_{\mathcal{X}/S}\otimes \mathcal{D} \otimes \mathcal{L})$ carries a smooth hermitian metric. In case $\mathcal{L}$ is relatively positive, we give an explicit formula for its curvature. The result applies to families of log-canonically polarized pairs. Moreover we show that it improves the general positivity result of Berndtsson-P\u{a}un in a special situation of a big line bundle.

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Philipp Naumann. 2020-05-04. Positivity of direct images with a Poincar\'e type twist. https://arxiv.org/abs/2005.01500

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