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

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 Kähler, 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.

37 pages, to appear in the Journal of Geometric Analysis (special issue "Proc. Conf. Complex Analysis, Geometry, and Dynamics III - Portorož 2024", dedicated to Josip Globevnik for his 80th birthday)