$\bar{\partial}$-equation on $(p,q)$-forms on conic neighbourhoods of $1$-convex manifolds

TL;DR

This paper develops a metric on vector bundles over conic neighborhoods of 1-convex manifolds, enabling analysis of the $ar{ abla}$-equation with polynomial poles and positive curvature in specific bidegrees.

Contribution

It introduces a new metric construction on vector bundles over conic neighborhoods of 1-convex manifolds with polynomial poles and positive Nakano curvature.

Findings

Constructed a metric on vector bundles with polynomial poles.

Achieved solutions to the $ar{ abla}$-equation in specified bidegrees.

Extended analysis of complex structures on 1-convex manifolds.

Abstract

We construct a metric on a vector bundle $E \rightarrow Z$ restricted to a conic neighbourhood of a relatively compact $1$-convex section of a submersion $Z \rightarrow X$ with at most polynomial poles at the boundary and positive Nakano curvature tensor in bidegree $(p,q).$