The Shafarevich conjecture on holomorphic convexity of universal covers

Surface manuscript, Theorem 1.1: a smooth connected projective complex surface $X$ containing a connected nodal curve $Z_0$ with smooth rational components such that $\pi_1(Z_0)\to\pi_1(X)$ has infinite image; a component of its preimage in $\widetilde X$ is a closed noncompact union of compact rational curves, so hulls are noncompact and $\widetilde X$ is not holomorphically convex. By the linear Shafarevich theorem $\pi_1(X)$ has no faithful finite-dimensional complex representation. Fourfold manuscript, Theorem 1.1: a smooth projective fourfold with large fundamental group (no positive-dimensional compact subvariety in $\widetilde X$) whose universal cover is not Stein, hence not holomorphically convex; this disproves the large-fundamental-group form as well. Neither paper addresses quasi-projective or Kahler variants beyond these.

Surface manuscript, Theorem 1.1: a smooth connected projective complex surface $X$ containing a connected nodal curve $Z_0$ with smooth rational components such that $\pi_1(Z_0)\to\pi_1(X)$ has infinite image; a component of its preimage in $\widetilde X$ is a closed noncompact union of compact rational curves, so hulls are noncompact and $\widetilde X$ is not holomorphically convex. By the linear Shafarevich theorem $\pi_1(X)$ has no faithful finite-dimensional complex representation. Fourfold manuscript, Theorem 1.1: a smooth projective fourfold with large fundamental group (no positive-dimensional compact subvariety in $\widetilde X$) whose universal cover is not Stein, hence not holomorphically convex; this disproves the large-fundamental-group form as well. Neither paper addresses quasi-projective or Kahler variants beyond these.