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Math-2566 · mathematics · Algebraic geometry

Topological right-triviality of constant-Milnor surface families

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let F(x,t) be holomorphic near ${0}×\Delta $ in $C^{3}×\Delta $, with $f_t(0)=0$ and an isolated critical point at zero in each fiber, and with constant Milnor number. After shrinking representatives and the disk, must there be a jointly continuous homeomorphism germ $\Phi (x,t)=(\varphi _t(x),t)$, with continuous inverse, $\varphi _t(0)=0$ and $\varphi _{0}=id$, satisfying $f_t(\varphi _t(x))=f_{0}(x)$?

Short English statement

Let F(x,t) be holomorphic near ${0}×\Delta $ in $C^{3}×\Delta $, with $f_t(0)=0$ and an isolated critical point at zero in each fiber, and with constant Milnor number. After shrinking representatives and the disk, must there be a jointly continuous homeomorphism germ $\Phi (x,t)=(\varphi _t(x),t)$, with continuous inverse, $\varphi _t(0)=0$ and $\varphi _{0}=id$, satisfying $f_t(\varphi _t(x))=f_{0}(x)$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2566-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2566

Manuscript 1 (family 064, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Topological right-triviality of constant-Milnor surface families

OpenAI's claim

A one-parameter isolated surface-hypersurface family with constant Milnor number is claimed to admit a jointly continuous ambient trivialization preserving both parameter and function value. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For a holomorphic family $F:(\mathbb C^3\times\Delta,\{0\}\times\Delta)\to(\mathbb C,0)$, put $f_t(x)=F(x,t)$. Assume each $f_t(0)=0$ has an isolated critical point and constant Milnor number \[ \mu(f_t,0)=\dim_{\mathbb C}\mathcal O_{\mathbb C^3,0}/(\partial_1f_t,\partial_2f_t,\partial_3f_t)=\mu(f_0,0). \] OpenAI claims, after shrinking the disk and representatives, a parameter-preserving homeomorphism germ $\Phi(x,t)=(\phi_t(x),t)$ with $\phi_t(0)=0$, $\phi_0=\mathrm{id}$ and $f_t(\phi_t(x))=f_0(x)$. Both $\Phi$ and its inverse are jointly continuous along the origin section. This is topological right-triviality, including defining functions rather than only zero sets.

Outline of the argument

Manuscript 1 (family 064, 2026-09-24): Semistable reduction and a log program compare extra boundary surfaces by compact-support Euler characteristics. Constant Milnor number cancels the main contribution, and nonnegative smoothing excesses plus the orbifold surface inequality force their log squares to vanish; resolution transport then extends to the ambient function levels.

Formal verification

Family 064 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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