Hyperbolicity of Moduli Spaces of Log-Canonically Polarized Manifolds

To, Wing-Keung; Yeung, Sai-Kee

doi:10.1007/s12220-020-00380-8

Cited by 1 publication

(3 citation statements)

References 26 publications

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“…A proof of the more general curvature formula, including the higher direct images, was claimed in [22]. However, the arguments in this work seem to be incomplete.…”

Section: Differential Geometric Setup and Statement Of Resultsmentioning

confidence: 88%

“…However, the arguments in this work seem to be incomplete. One major issue is the smooth dependency of the fibrewise harmonic projections without the usual assumption of having constant dimension for the space of fibrewise harmonic forms; compare to [22, Lemma 2]. This is impossible given that linear independency is an open property; compare to the argument in the proof of [15, Lemma 7.7].…”

Section: Differential Geometric Setup and Statement Of Resultsmentioning

confidence: 99%

“…This is impossible given that linear independency is an open property; compare to the argument in the proof of [15, Lemma 7.7]. Moreover, it is unclear how to derive the square integrability of some Lie-differentiated forms from the arguments given in [22, pp. 2958f], which is another key ingredient in the computation.…”

Section: Differential Geometric Setup and Statement Of Resultsmentioning

confidence: 99%

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Positivity of direct images with a Poincaré type twist

Naumann

2022

Forum of Mathematics, Sigma

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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.

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“…A proof of the more general curvature formula, including the higher direct images, was claimed in [22]. However, the arguments in this work seem to be incomplete.…”

Section: Differential Geometric Setup and Statement Of Resultsmentioning

confidence: 88%

Section: Differential Geometric Setup and Statement Of Resultsmentioning

confidence: 99%