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Hyperplane arrangements with non-formal Milnor fibers

Alexander I. Suciu

Abstract

Each complex hyperplane arrangement $\mathcal{A}$ gives rise to a Milnor fibration of its complement. Building on work of Zuber, we give a combinatorial sufficient condition for the Milnor fiber $F(\mathcal{A})$ to be non-$1$-formal, expressed in terms of the multinet structure on $\mathcal{A}$, and use it to produce an infinite family of monomial arrangements $\mathcal{A}(3k,3k,3)$ with non-formal Milnor fibers. We also review the relevant background on cohomology jump loci, formality, and the topology of Milnor fibers of arrangements.

Hyperplane arrangements with non-formal Milnor fibers

Abstract

Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Building on work of Zuber, we give a combinatorial sufficient condition for the Milnor fiber to be non--formal, expressed in terms of the multinet structure on , and use it to produce an infinite family of monomial arrangements with non-formal Milnor fibers. We also review the relevant background on cohomology jump loci, formality, and the topology of Milnor fibers of arrangements.
Paper Structure (31 sections, 15 theorems, 36 equations, 1 figure)

This paper contains 31 sections, 15 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cohomology jump loci and formality
  3. Characteristic and resonance varieties
  4. Quasi-projective varieties
  5. Formality and the Tangent Cone Theorem
  6. Hyperplane arrangements
  7. Complements and intersection lattices
  8. The Orlik--Solomon algebra and formality
  9. Cohomology jump loci of arrangements
  10. Multinets and resonance components
  11. Milnor fibrations of arrangements
  12. The Milnor fibration
  13. Betti numbers and algebraic monodromy
  14. Pencils of lines in the plane
  15. The reduced diagonal character
  16. ...and 16 more sections

Key Result

Theorem 1.2

Let $\mathcal{A}$ be a central arrangement of $n$ hyperplanes in $\mathbb{C}^\ell$ supporting at least two distinct reduced $3$-multinets. Then the Milnor fiber $F(\mathcal{A})$ is not $1$-formal.

Figures (1)

  • Figure 1: The pincer argument: two net components $T_1,T_2$ of $\mathcal{V}^1_1(U)$ meet at the torsion point $\rho_n$, forcing $\dim H^{1,0}(F)\ge 2$ (red), while the lifted pencil forces $\dim(E\cap H^{1,0}(F))=1$ (blue). These are incompatible with $1$-formality via the Tangent Cone Theorem.

Theorems & Definitions (23)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Budur--Wang BW
  • Theorem 2.2: Arapura Ar
  • Theorem 2.3: ACM
  • Theorem 2.4: Măcinic Mc10
  • Theorem 2.5: Tangent Cone Theorem DPS-dukeDP-ccm
  • Lemma 2.6: DP-pisa
  • Proposition 3.1
  • Definition 3.2
  • ...and 13 more