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Tóth-Varolin Conjecture

The Tóth-Varolin conjecture asserts that a Stein manifold diffeomorphic to C^n and possessing the holomorphic density property is biholomorphic to C^n. A biholomorphic map, rather than an algebraic isomorphism, is required.

Gorai (2026) reported a counterexample given by the smooth affine fourfold

X={(zeta,z_0,z_1,z_2,z_3) in C^5:zeta^6z_0^7+(1+z_1zeta)z_2+zetaz_3^2=1}.

The asserted properties are that X is diffeomorphic to C^4, has the holomorphic density property, and is not biholomorphic to C^4. The last assertion is stronger than failure of algebraic isomorphism. The paper is labeled preliminary.

See also

Biholomorphic Map, Holomorphic Density Property, Stein Manifold

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References

Gorai, S. "A New Class of Smooth Algebraic Varieties with the Density Property and a Counterexample to the Tóth-Varolin Conjecture." 20 Oct 2026. https://arxiv.org/abs/2610.27660.

Cite this as:

Weisstein, Eric W. "Tóth-Varolin Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Toth-VarolinConjecture.html

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