Positivity of direct images with a Poincar\'e type twist
- Resource Type
- Working Paper
- Authors
- Naumann, Philipp
- Source
- Subject
- Mathematics - Complex Variables
Mathematics - Algebraic Geometry
Mathematics - Differential Geometry
- Language
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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