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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, and its proposed biholomorphic obstruction has not been independently verified here.

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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