Isolated Hypersurface Singularities May Be Stably Degenerate

Technical Summary: Isolated Hypersurface Singularities May Be Stably Degenerate

Problem Statement
The paper addresses a long-standing question in singularity theory originally posed by Arnol'd in 1975: Does every isolated hypersurface singularity admit a stable equivalence to a Newton non-degenerate polynomial? A polynomial is Newton non-degenerate if its restriction to the torus $(\mathbb{C}^*)^N$ has no critical points on any compact face of its Newton polyhedron. While it is known that singularities can be degenerate in any fixed coordinate system, the question remains whether stabilization (adding non-degenerate quadratic forms in new variables) can always eliminate degeneracy. The authors prove the existence of counterexamples to this conjecture.

Methodology
The proof is an existence argument relying on the interplay between the mixed Hodge structure of contact loci and the geometry of moduli spaces of abelian varieties. The core strategy involves constructing a specific singularity $f$ and demonstrating that its stable invariants cannot arise from any family of Newton non-degenerate singularities.

1. Construction of the Singularity:
   The authors construct a polynomial $f = F + h$, where:

   • $F$ is a homogeneous polynomial of degree $d$ defining a hypersurface in projective space that is birational to a "sufficiently general" principally polarized abelian variety $A$ of high dimension ($g \geq 16$).
   • $h$ is a polynomial of degree strictly greater than $d$ (e.g., of Fermat type) ensuring $f$ has an isolated singularity at the origin.
   • The specific abelian variety $A$ is chosen to satisfy three properties (Lemma 3.3): it is not contained in any rational subvariety of its moduli space, its isogeny class contains no variety that is the generic fiber of a morphism defined over $\mathbb{Q}$ with constant Hodge structure, and its Neron-Severi group has rank 1.

2. Contact Loci and Mixed Hodge Structures:
   The authors analyze the contact locus $\chi_f$, the space of $d$-jets of curves tangent to the singularity. They compute the mixed Hodge structure (MHS) of the compactly supported cohomology of the contact locus of a stabilization $f_k$.

   • A key result (Lemma 6.10) establishes that a specific weight-graded piece of the MHS of the contact locus of $f_k$ is isomorphic to a Tate twist of the cohomology of the abelian variety $A$ (specifically $H^{2g-1}(A; \mathbb{Q})(-m)$).

3. Contradiction via Moduli Spaces:
   The proof proceeds by contradiction. Suppose a stabilization of $f$ were Newton non-degenerate after a coordinate change.

   • The space of Newton non-degenerate polynomials with a fixed Newton polyhedron forms a family parameterized by a rational variety $T$.
   • For such families, the cohomology of the contact locus forms a locally constant sheaf (Lemma 5.8), inducing a period map from the parameter space $T$ to the moduli space of polarized abelian varieties $A_{g, \delta}$.
   • Since $T$ is rational and the moduli space $A_{g, \delta}$ (for $g \geq 16$) is of general type (and thus contains no rational curves passing through a sufficiently general point), any algebraic map from $T$ to $A_{g, \delta}$ must be constant.
   • However, the specific abelian variety $A$ constructed was chosen to lie outside the image of any such constant map defined over $\mathbb{Q}$ (Lemma 3.3). This contradiction implies that no coordinate change can render the stabilization Newton non-degenerate.

Key Contributions and Results

• Theorem 1.1: The existence of an isolated hypersurface singularity that is stably degenerate is proven. This provides a negative answer to Arnol'd's 1975 problem.
• Geometric Characterization: The paper identifies that the obstruction to stable non-degeneracy lies in the global geometric properties of the moduli space of abelian varieties. Specifically, the "non-uniruledness" (being of general type) of high-dimensional moduli spaces prevents certain Hodge structures from being realized by families of Newton non-degenerate singularities.
• Algorithmic Existence: While the primary proof is non-constructive, Section 8 outlines a theoretical algorithm to produce an explicit example. This involves using Gröbner basis techniques to find a point in the moduli space of abelian varieties with algebraically independent coordinates, ensuring the resulting singularity satisfies the necessary constraints. The authors note this is computationally infeasible in practice due to the doubly exponential complexity and the massive degree of the resulting polynomials.

Significance
The paper resolves a fundamental open problem in singularity theory by demonstrating that the class of Newton non-degenerate singularities is not stable-equivalent to the class of all isolated hypersurface singularities. The significance of the result is twofold:

1. Theoretical: It establishes a deep connection between the local analytic properties of singularities and the global geometry of moduli spaces of abelian varieties.
2. Methodological: It introduces the use of mixed Hodge structures on contact loci as a powerful invariant for distinguishing stable equivalence classes, specifically leveraging the rigidity of period maps on moduli spaces of general type.

The authors note that their argument relies on the existence of abelian varieties of dimension $g \geq 16$ with specific properties, and they reference Stevens' work suggesting that such singularities might be generic within certain strata of constant Milnor number. The paper concludes by suggesting that combining these methods with Floer theory could potentially strengthen the result to show that such singularities cannot be deformed into Newton non-degenerate ones within a $\mu$-constant family.