The Shafarevich holomorphic convexity conjecture asserts that the universal cover of every smooth connected projective complex variety, with its lifted complex structure, is holomorphically convex. For a variety with large fundamental group this predicts that the universal cover is Stein, because it has no positive-dimensional compact analytic subvariety.
Shafarevich Holomorphic Convexity Conjecture
See also
Complex Surface, Holomorphic Convexity, Large Fundamental Group, Projective Variety, Stein Manifold, Universal CoverExplore with Wolfram|Alpha
References
Deng, Y. "Topology, Hyperbolicity, and the Shafarevich Conjecture for Complex Algebraic Varieties." 30 Dec 2025. https://arxiv.org/abs/2512.24458.Cite this as:
Weisstein, Eric W. "Shafarevich Holomorphic Convexity Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ShafarevichHolomorphicConvexityConjecture.html