Geodesic
Positivity of direct images with a Poincaré type twist
Forum of Mathematics, Sigma, Tome 10 (2022)

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We consider a holomorphic family $f:\mathscr {X} \to S$ of compact complex manifolds and a line bundle $\mathscr {L}\to \mathscr {X}$. Given that $\mathscr {L}^{-1}$ carries a singular hermitian metric that has Poincaré type singularities along a relative snc divisor $\mathscr {D}$, the direct image $f_*(K_{\mathscr {X}/S}\otimes \mathscr {D} \otimes \mathscr {L})$ carries a smooth hermitian metric. If $\mathscr {L}$ 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-Păun in a special situation of a big line bundle. 
@article{10_1017_fms_2022_79,
     author = {Philipp Naumann},
     title = {Positivity of direct images with a {Poincar\'e} type twist},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {10},
     year = {2022},
     doi = {10.1017/fms.2022.79},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.79/}
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Philipp Naumann. Positivity of direct images with a Poincaré type twist. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.79

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