On torsion of \(\bar\partial\)-closed currents on complex analytic spaces (Q2712603)

Let \(X\) be a reduced complex analytic space and \({\mathcal O}_X\) be the sheaf of germs of holomorphic functions on \(X\). It is shown that the cohomology of the Dolbeault-Grothendieck complex of smooth forms on \(X\) can be not \({\mathcal O}_X\)-coherent, and that there can exist \(\overline\partial\)-closed currents of type \((r,0)\) with support in the set of singular points of \(X\). The two statements are proved by considering the curve \(\{(s^k+ s^{k+1}, s^{k+2}): s\in\mathbb{C}\}\), \(k\geq 4\).