Links of singularities of inner non-degenerate mixed functions

TL;DR

The paper extends the Newton-boundary framework from holomorphic to mixed polynomials $f:\mathbb{C}^2\to\mathbb{C}$, introducing (strong) inner non-degeneracy and a niceness condition to control singularities. It provides explicit, braid-based descriptions of the links of singularities, showing that under inner non-degeneracy the link is determined by the Newton boundary, and that strong inner non-degeneracy yields isolated singularities with links given by nested braid closures. The authors develop a semi-holomorphic principal parts approach to realize real algebraic links, then generalize to non-semi-holomorphic faces, culminating in a new method to construct real algebraic links via P-fibered braids and $O$-multiplicity. This yields an infinite family of real algebraic links in $S^3$, expanding the known landscape of real singularity topology and clarifying how Newton data governs link topology in the mixed setting.

Abstract

We introduce the notion of a (strongly) inner non-degenerate mixed function $f:\mathbb{C}^2\to\mathbb{C}$. We show that inner non-degenerate mixed polynomials have weakly isolated singularities and strongly inner non-degenerate mixed polynomials have isolated singularities. Furthermore, under one additional assumption, which we call "niceness", the links of these singularities can be completely characterized in terms of the Newton boundary of $f$. In particular, adding terms above the Newton boundary does not affect the topology of the link. This allows us to define an infinite family of real algebraic links in the 3-sphere.

Figures (4)

• Figure 1: The Newton polygon $\Gamma_+(f)$ of a boundary polynomial $f$.
• Figure 2: a) The diagram of a braid. b) The diagram of the closure of a braid. c) The union of a closure of a braid and its braid axis.
• Figure 3: In the top row: a braid $B_{1}$ and two affine braids $B_{2},B_{3}$. In the bottom row: the braids $B(B_1,B_2)$ and $B(B_1,B_2,B_3)$.
• Figure 4: In the top row: links $L_1$ in $\mathbb{C}\times S^1$, $L_2$ in $(\mathbb{C}\backslash\{0\})\times S^1$, and $L_3$ in the complementary torus $S^1\times\mathbb{C}$. In the bottom row: the links $[L_1,L_2]$ and $L([L_1,L_2],L_3)$.

Theorems & Definitions (47)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Example 2.3
• Definition 2.4: Oka Oka2010
• Remark 2.5
• Definition 2.6
• Definition 3.1