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A Miyaoka-Yau inequality for hyperplane arrangements in $\mathbb{CP}^n$

Martin de Borbon, Dmitri Panov

Abstract

Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{CP}^n$. We define a quadratic form $Q$ on $\mathbb{R}^{\mathcal{H}}$ that is entirely determined by the intersection poset of $\mathcal{H}$. Using the Bogomolov-Gieseker inequality for parabolic bundles, we show that if $\mathbf{a} \in \mathbb{R}^{\mathcal{H}}$ is such that the weighted arrangement $(\mathcal{H}, \mathbf{a})$ is stable, then $Q(\mathbf{a}) \leq 0$. As an application, we consider the symmetric case where all the weights are equal. The inequality $Q(a, \ldots, a) \leq 0$ gives a lower bound for the total sum of multiplicities of codimension $2$ intersection subspaces of $\mathcal{H}$. The lower bound is attained when every $H \in \mathcal{H}$ intersects all the other members of $\mathcal{H} \setminus \{H\}$ along $(1-2/(n+1))|\mathcal{H}| + 1$ codimension $2$ subspaces; extending from $n=2$ to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane.

A Miyaoka-Yau inequality for hyperplane arrangements in $\mathbb{CP}^n$

Abstract

Let be a hyperplane arrangement in . We define a quadratic form on that is entirely determined by the intersection poset of . Using the Bogomolov-Gieseker inequality for parabolic bundles, we show that if is such that the weighted arrangement is stable, then . As an application, we consider the symmetric case where all the weights are equal. The inequality gives a lower bound for the total sum of multiplicities of codimension intersection subspaces of . The lower bound is attained when every intersects all the other members of along codimension subspaces; extending from to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane.

Paper Structure

This paper contains 63 sections, 132 theorems, 438 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main result: klt and CY arrangements
  3. Sketch proof of Theorem \ref{['thm:main']}
  4. Context: PK manifolds
  5. Outline
  6. Hyperplane arrangements
  7. Basic definitions
  8. Irreducible arrangements and subspaces
  9. Weighted arrangements
  10. The resolution
  11. The minimal De Concini-Procesi model
  12. Construction of $X$ as an iterated blowup
  13. Nested subsets of $\mathcal{L}_{\textup{irr}}$ and intersections of divisors $D_L$
  14. Example: tetrahedron in $\mathbb{CP}^3$
  15. Picard group and cohomology of X
  16. ...and 48 more sections

Key Result

Theorem 1.1

Suppose that $(\mathcal{H}, \mathbf{a})$ is a weighted arrangement that is both klt and CY. Then the following inequality holds:

Figures (3)

  • Figure 1: The arrangement $\mathcal{H}$ of $10$ planes in $\mathbb{CP}^3$ spanned by triplets of $5$ points in general linear position.
  • Figure 2: Reducible (red) and irreducible (green) intersections.
  • Figure 3: The Non-Pappus matroid.

Theorems & Definitions (404)

  • Remark
  • Remark
  • Remark
  • Theorem 1.1
  • Remark 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • ...and 394 more