Holomorphic $p$-Contact and $s$-Symplectic Line Bundles

TL;DR

This paper extends the concepts of holomorphic $p$-contact and $s$-symplectic structures to line bundle-valued cases on compact complex manifolds, exploring their properties and potential for projectivity, unlike their scalar versions.

Contribution

It introduces line bundle-valued holomorphic $p$-contact and $s$-symplectic structures and studies their geometric properties, including the positivity of the canonical bundle.

Findings

Holomorphic $p$-contact and $s$-symplectic structures can be projective.

The canonical bundle's positivity properties are analyzed with singular Hermitian metrics.

Scalar versions of these structures are never Kähler.

Abstract

We generalise the notions of scalar-valued holomorphic $p$-contact and $s$-symplectic structures introduced recently on compact complex manifolds by the second-named author jointly with H. Kasuya and L. Ugarte to their analogues with values in a holomorphic line bundle. We then study the resulting holomorphic $p$-contact and $s$-symplectic manifolds which, unlike their scalar counterparts that are never K\"ahler, can even be projective. In particular, we investigate the (lack of) positivity properties of the canonical bundle of these manifolds when it is given a possibly singular Hermitian fibre metric. One of the tools used is a very recent regularisation result for $m$-psh functions obtained jointly by S. Dinew and the second-named author.