Toric reduction of singularities for Newton nondegenerate $p$-forms

TL;DR

This work develops a toric-geometric framework to study singularities of holomorphic $p$-forms through Newton polyhedra. By extending the Newton polyhedron construction to $p$-forms and introducing Newton nondegeneracy (NND), it characterizes when a single toric morphism suffices to reduce singularities, via a regular refinement of the dual fan and the pull-back expression $\pi^{*}\eta(Y)=Y^T(\sum_{|K|=p} \overline{f}_K(Y) \frac{dY_K}{Y_K})$. A key result is that $\eta$ is NND if and only if the coefficients $\overline{f}_K(Y)$ have no common zeros on every affine chart, providing a concrete toric criterion for resolution. The authors then apply this to $(n-1)$-forms, showing that Newton nondegeneracy ensures the strict transform has at most nonnilpotent singularities, yielding a toric-resolution-type conclusion without integrability requirements. Overall, the paper connects Newton polyhedron data to toric reduction, offering a single-morphism, combinatorially driven approach to singularities of foliations and related differential forms.

Abstract

We study a class of holomorphic $p$-forms satisfying nondegeneracy conditions expressed through their Newton polyhedron and called Newton nondegenerate (NND). We give a characterization of NND $p$-forms by their toric reduction of singularities defined through a regular refinement of their dual fan. We then present an application of this result to the study of singularities of $(n-1)$-forms on $\mathbb{C}^n$.

Figures (2)

• Figure 1: Newton polyhedron of $\eta$.
• Figure 2: Newton boundary of $\Gamma(\omega)$.

Theorems & Definitions (20)

• Theorem
• Theorem
• Definition 1.1
• Example 1.2
• Example 1.3
• Definition 1.4
• Definition 1.5
• Remark 1.6
• Example 1.7
• Theorem 2.1