Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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F. Fagioli

A note on Griffiths' conjecture about the positivity of Chern-Weil forms

created by fagioli on 09 Feb 2022


Published Paper

Inserted: 9 feb 2022
Last Updated: 9 feb 2022

Journal: Differ. Geom. Appl.
Volume: 81
Year: 2022
Doi: 10.1016/j.difgeo.2022.101848

ArXiv: 2012.12815 PDF
Links: Differ. Geom. Appl.


Let $ (E,h) $ be a Griffiths semipositive Hermitian holomorphic vector bundle of rank $ 3 $ over a complex manifold. In this paper, we prove the positivity of the characteristic differential form $ c_1(E,h) \wedge c_2(E,h) - c_3(E,h) $, thus providing a new evidence towards a conjecture by Griffiths about the positivity of the Schur polynomials in the Chern forms of Griffiths semipositive vector bundles. As a consequence, we establish a new chain of inequalities between Chern forms. Moreover, we point out how to obtain the positivity of the second Chern form $ c_2(E,h) $ in any rank, starting from the well-known positivity of such form if $ (E,h) $ is just Griffiths positive of rank $ 2 $. The final part of the paper gives an overview on the state of the art of Griffiths' conjecture, collecting several remarks and open questions.

Keywords: Chern-Weil forms, Griffiths' conjecture, Flag bundles, Push-forward formulae, Schur forms

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