Inner Metric Geometry of Complex Algebraic Surfaces with Isolated Singularities

We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to the inner metric, to cones. The technique used to prove the nonexistence of the metric conic structure is related to a development of Metric Homology. The class of the examples is rather large and it includes some surfaces of Brieskorn.

💡 Research Summary

The paper investigates the inner‑metric geometry of complex algebraic surfaces that possess isolated singularities and demonstrates that a large class of such singularities fail to be metrically conic. In the classical setting, singular points of complex varieties are often studied via their ambient Euclidean metric; under that viewpoint the germ of a singularity is topologically a cone, and many analytic invariants are built on this conical approximation. The authors, however, focus on the intrinsic metric defined by the length of curves lying entirely on the surface (the inner metric). They ask whether the germ ((X,0)) of a surface with an isolated singularity is bi‑Lipschitz equivalent, with respect to this inner metric, to a genuine metric cone.

To answer this, they introduce a new homological tool called Metric Homology. Metric Homology refines ordinary singular homology by imposing a scale‑dependent condition: a chain is admissible only if it can be realized inside a ball of radius (\epsilon) and, crucially, if it cannot be contracted to a point within a proportionally smaller ball. The existence of a non‑shrinkable cycle in a given dimension yields a non‑trivial metric homology group, which in turn obstructs the existence of a bi‑Lipschitz cone model. In a genuine metric cone every cycle can be arbitrarily shrunk by scaling, so non‑trivial metric homology is incompatible with conicity.

Armed with this framework, the authors construct explicit families of surfaces that violate the conic condition. The first family consists of Brieskorn surfaces
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