\({\bar\partial}\)-problem on a family of weakly pseudoconvex manifolds (Q1080000)

Let \(E\in Pic^ 0(T^ n)\) be a holomorphic line bundle on the complex torus \(T^ n\) with Chern class zero. The authors study the cohomology group \(H^ p(E,0)\) of the structure sheaf 0 of E. Let d(, ) be the invariant distance on \(Pic^ 0(T^ n)\), then the main result is the following criterion: (1) If \(d(I,E^{\ell})=0\) for some \(\ell \geq 1\), then \(H^ p(E,0)\) is an infinite-dimensional Hausdorff space (1\(\leq p\leq n);\) (2) If there exists \(a>0\) such that \(\exp (-a\ell)\leq d(I,E^{\ell})\) for any \(\ell \geq 1\), then dim \(H^ p(E,0)=\left( \begin{matrix} n\\ p\end{matrix} \right)\) (1\(\leq p\leq n);\) (3) If \(d(I,E^{\ell})\neq 0\) for any \(\ell \geq 1\) and \(\lim \inf_{\ell \to \infty}\exp (a\ell)d(I,E^{\ell})=0\) for any \(a>0\), then \(H^ p(E,0)\) is not Hausdorff (1\(\leq p\leq n).\) The complete proof will appear elsewhere.