Metric Properties of Parabolic Ample Bundles

Abstract We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern–Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern cla...

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Bibliographic Details
Published in:International mathematics research notices 2020-11, Vol.2020 (23), p.9336-9369
Main Authors: Biswas, Indranil, Pingali, Vamsi Pritham
Format: Article
Language:English
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Summary:Abstract We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern–Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favor for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type result holds and that there are metrics on stable bundles over surfaces whose Schur forms are positive.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rny259