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

Abstract:

Let $X$ be a $1$-convex manifold with the exceptional set $S$, which is also a manifold, $Z$ a complex manifold, $Z \rightarrow X$ a holomorphic submersion, $a: X \rightarrow Z$ a holomorphic section and $S \subset U \Subset X$ an open relatively compact $1$-convex set. We construct a metric on a vector bundle $E \rightarrow Z$ restricted to a neighbourhood $V$ of $a(U),$ conic along $a(S)$ with at most polynomial poles at $a(S)$ and positive Nakano curvature tensor in bidegree $(p,q).$