For every holomorphic map germ boldsymboln:(C^2,0)→(C^3,0) that admits an injective representative, the differential at the origin is nonzero: (d boldsymboln)_0≠ 0. Equivalently, there is no injective holomorphic map germ boldsymboln:(C^2,0)→(C^3,0) with (d boldsymboln)_0=0.

Lê’s conjecture

Lê’s conjecture concerns the structure of injective holomorphic map germs and related complex-surface singularities. Earlier literature records it as open, with only partial and conditional results.

Known results

The two-dimensional formulation relates injective analytic germs from $(\mathbb{C}^{2},0)$ to $(\mathbb{C}^{3},0)$ to smooth singular loci and constant transversal Milnor number (2006).
Conditional equisingularity results were obtained assuming Lê’s conjecture, with some unconditional results under additional smooth-branch hypotheses (2006).
A 2019 survey explicitly described the conjecture as still open and having only partial results.

October 2026 reduction

Pablo Portilla Cuadrado’s new paper is reported to reduce the remaining problem, using plane-curve singularities, cobordisms, and knot Floer bounds, to excluding orders $2$ , $3$ , and $4$ . This is a claimed advance, but the supplied evidence does not independently verify the reduction.

Current status (as of October 2026): Lê’s conjecture remains open; the reported reduction to orders $2$ , $3$ , and $4$ is unverified, and those cases remain to be excluded.

Lê’s conjecture

Primary source

Additional references

Progress summary

Solutions 0