On a construction of hermitian metrics on holomorphic vector bundles

Abstract

With any holomorphic vector bundle $E$ over a compact base one can associate a line bundle, denoted $O_{PE}(1)$. According to a conjecture of Griffiths, if $O_{PE}(1)$ admits a positively curved hermitian metric $k$, then $E$ also admits a positively curved hermitian metric, $h$. In this paper we show that, while the conjecture may be correct, it is not possible to obtain $h$ out of $k$ by a fiberwise construction that is functorial.

The abstract already appeared, by mistake, accompanying submission arXiv:2608.22689. The mistake has since been corrected; this abstract here belongs to the current submission