Hirschowitz's formal principle conjecture for smooth unbendable rational curves

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Let AA be a smooth unbendable rational curve in a complex manifold XX. The curve AA satisfies the formal principle: whenever a compact complex submanifold A~\widetilde{A} in a complex manifold X~\widetilde{X} admits a formal isomorphism

ψ:(A/X)∞⟶≅(A~/X~)∞,\psi:(A/X)_{\infty}\stackrel{\cong}{\longrightarrow}(\widetilde{A}/\widetilde{X})_{\infty},

there are neighborhoods UU of AA in XX and U~\widetilde{U} of A~\widetilde{A} in X~\widetilde{X} and a biholomorphic map Ψ:U⟶≅U~\Psi:U\stackrel{\cong}{\longrightarrow}\widetilde{U} such that Ψ(A)=A~\Psi(A)=\widetilde{A}. Hirschowitz's formal principle conjecture. A smooth unbendable rational curve satisfies the formal principle. This is presented as a special case of Hirschowitz's conjecture on the formal principle; the supplied text does not state whether it has been resolved.

References

Primary source

Jun-Muk Hwang, “Geometry of Neighborhoods of Minimal Rational Curves”, arXiv:2605.24303 (2026).

Additional references

2 papers in this index state this conjecture (2019–2026). The statement above is taken from the most recent of them; the others are arXiv:1903.09490.

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