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Signature of the Milnor fiber of parametrized surfaces

R. Giménez Conejero, Gergő Pintér

Abstract

We compute the signature of the Milnor fiber of certain type of non-isolated complex surface singularities, namely, images of finitely determined holomorphic germs. An explicit formula is given in algebraic terms. As a corollary we show that the signature of the Milnor fiber is a topological invariant for these singularities. The proof combines complex analytic and smooth topological techniques. The main tools are Thom-Mather theory of map germs and the Ekholm-Szűcs-Takase-Saeki formula for immersions. We give a table with many examples for which the signature is computed using our formula.

Signature of the Milnor fiber of parametrized surfaces

Abstract

We compute the signature of the Milnor fiber of certain type of non-isolated complex surface singularities, namely, images of finitely determined holomorphic germs. An explicit formula is given in algebraic terms. As a corollary we show that the signature of the Milnor fiber is a topological invariant for these singularities. The proof combines complex analytic and smooth topological techniques. The main tools are Thom-Mather theory of map germs and the Ekholm-Szűcs-Takase-Saeki formula for immersions. We give a table with many examples for which the signature is computed using our formula.
Paper Structure (31 sections, 26 theorems, 154 equations, 7 figures, 1 table)

This paper contains 31 sections, 26 theorems, 154 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary of the result
  3. The context of the result
  4. Structure of the article
  5. Acknowledgments
  6. Preliminaries
  7. Map germs and invariants
  8. The associated immersion
  9. The image and the double points
  10. The boundary of the Milnor fiber
  11. The surgery providing $\partial F_f$
  12. Computation of the vertical indices
  13. Slice singular manifolds of immersions
  14. Slice singular manifold of the associated immersion
  15. Slice singular manifolds for the boundary of the Milnor fiber
  16. ...and 16 more sections

Key Result

Theorem 1.1

Consider a finitely determined holomorphic germ $f: (\mathbb C^2, 0) \to (\mathbb C^3, 0)$. Let $(\mathcal{S}, 0)=({\rm Im}(f), 0)$ be its image and $F_f$ its Milnor fiber. Then, its signature $\sigma(F_f)$ satisfies

Figures (7)

  • Figure 1: Stable singularities from $\mathbb{C}^2$ to $\mathbb{C}^3$.
  • Figure 2: An example of a map with a twisted component (left, cf. \ref{['ex:cc', 'ex:s0']}) and another with an untwisted component (right, cf. $S_1$ in \ref{['ex:sk']}). The associated immersion $f|_{\mathbb{S}^3}: f^{-1}(\mathbb{S}^5_{\epsilon}) \cong \mathbb{S}^3 \to \mathbb{S}^5_{\epsilon}$; the double point sets $\bm{D}$ and $f(\bm{D})$; and the link $\bm{\delta}$ of $\bm{D}$, coinciding with the double point set of $f|_{\mathbb{S}^3}$ in the source.
  • Figure 3: Two representations of the pushing out $\Delta'$ of the generalized cross-caps $\Delta$ from the set of double values $m(D)$ of $m(M^4)$ (cf. $\ell(m)$ in \ref{['def:ell']}).
  • Figure 4: An $\mathscr{A}$-finite map germ $f$; a holomorphic stable deformation $f_s$ with complex cross-caps and triple values; a $\mathcal{C}^{\infty}$ stable deformation $f_s^\mathbb{R}$ with curves of generalized real cross-caps; and the map $h$ we construct, with new curves of generalized real cross-caps. Note that $f_s^\mathbb{R}$ and $h$ are slice singular manifolds of the associated immersion $f|_{\mathbb{S}^3}$ and of $\partial F \subset \mathbb{S}^5$, respectively. See \ref{['ss:assoc', 'ss:constructionh']}.
  • Figure 5: The homological cycle $\mathcal{C}_{f\{i\}}$ in $X$ corresponding to a couple of untwisted components $D_i$ and $D_{i'}$, cf. \ref{["eq:Cii'"]}. The cylinder (dashed) connecting them can be simplified to an identification of the boundaries (cf. \ref{['eq:def of X', 'eq:def of X2']}).
  • ...and 2 more figures

Theorems & Definitions (66)

  • Theorem 1.1: see \ref{['thm:main']}
  • Corollary 1.2: see \ref{['cor:signature top']}
  • Conjecture 1.3
  • Example 2.1: Complex Whitney umbrella, cross-cap
  • Definition 2.2
  • Lemma 2.3: see Mond2020
  • Lemma 2.5: cf. Nemethi2018
  • proof
  • Theorem 2.6
  • Definition 2.7
  • ...and 56 more