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