[Paper Review] Hypersurface complements, Milnor fibers and minimality of arrangements

Alexandru Dimca|ArXiv.org|Nov 27, 2000

Advanced Combinatorial Mathematics11 references3 citations

TL;DR

This paper establishes a topological link between the degree of the gradient map of a homogeneous polynomial and the Betti numbers of its hypersurface complement, proving that the $n$-th Betti number of the complement of a hyperplane arrangement equals the degree of the gradient map. Using Morse theory and polar curves, it shows that the complement is minimal and provides a homotopy-theoretic description of Milnor fibers, extending results on arrangements and their topology via Euler characteristic and constructible invariants.

ABSTRACT

We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. The main tools are polar curves and the affine Lefschetz theory developped by H. Hamm and A. Némethi. In the special case of the hyperplane arrangements, we strengthen some results due to Orlik and Terao (see Math. Ann. 301(1995)) and obtain an independant proof for the minimality of hyperplane arrangements (see Randell math.AT/0011101 for another proof of this result).

Motivation & Objective

• To relate the degree of the gradient map of a homogeneous polynomial to the topology of its hypersurface complement.
• To establish the minimality of hyperplane arrangement complements using homotopy type and Betti number equalities.
• To extend Orlik-Terao's results on Milnor fibers by proving that the number of $n$-cells attached in the homotopy type is exactly the number of critical points of a Morse function on the fiber.
• To provide a topological characterization of essential arrangements via the non-vanishing of the $n$-th Betti number.

Proposed method

• Uses the gradient map $\mathrm{grad}(h) : D(h) \to \mathbb{P}^n$ for a reduced homogeneous polynomial $h$ to relate its degree to the Euler characteristic of $D(h) \setminus H$, where $H$ is a generic hyperplane.
• Applies polar curves to compute the number of $n$-cells attached to $D(h) \cap H$ to obtain $\deg(\mathrm{grad}(h)) = (-1)^n \chi(D(h) \setminus H)$.
• Employs Morse theory on the Milnor fiber $F = \{f = 1\}$ via the function $g(x) = Q(x)\overline{Q}(x)$, showing it is a Morse function with $|C(g)|$ critical points.
• Uses the Affine Lefschetz Theorem and induction on the codimension to prove that the complement $M = D(Q)$ has the homotopy type of a CW complex with $b_k(M)$ $k$-cells for all $k$, establishing minimality.
• Applies the additivity of the Euler characteristic and Betti numbers via deletion-restriction techniques to compute $\chi(D(h))$ as an alternating sum of degrees of gradient maps on restrictions.
• Leverages the $a_h$-regularity condition and stratification theory to ensure transversality and control the limit behavior of sequences in the critical set.

Experimental results

Research questions

• RQ1 How is the degree of the gradient map $\mathrm{grad}(h)$ related to the Betti numbers of the complement $D(h)$ of a hypersurface defined by a reduced homogeneous polynomial $h$?
• RQ2 Under what conditions is the complement of a hyperplane arrangement minimal in the sense of having a CW structure with exactly $b_k$ $k$-cells for each $k$?
• RQ3 Can the number of $n$-cells attached to the Milnor fiber $F$ via Morse theory be precisely determined by the critical points of the function $g(x) = Q(x)\overline{Q}(x)$?
• RQ4 What topological condition on an arrangement ensures that $b_n(D(Q)) > 0$, and how does this relate to the essentiality of the arrangement?
• RQ5 How does the weight equivariant Euler polynomial of the Milnor fiber relate to the combinatorics of the arrangement lattice?

Key findings

• The $n$-th Betti number of the complement $D(Q)$ of a hyperplane arrangement equals the degree of the gradient map $\mathrm{grad}(Q)$, i.e., $b_n(D(Q)) = \deg(\mathrm{grad}(Q))$.
• The complement $M = D(Q)$ is minimal: it has the homotopy type of a CW complex where the number of $k$-cells equals the $k$-th Betti number for all $k \in \mathbb{N}$.
• The Milnor fiber $F$ of a generic homogeneous polynomial $f$ is homotopy equivalent to a CW complex obtained by attaching $|C(g)|$ $n$-cells to $F \cap N$, where $C(g)$ is the critical set of the Morse function $g(x) = Q(x)\overline{Q}(x)$.
• The arrangement is essential if and only if $\deg(\mathrm{grad}(Q)) > 0$, which is equivalent to $b_n(D(Q)) > 0$, and also equivalent to $\mathrm{grad}(Q)$ being dominant.
• For any constructible set in $\mathbb{P}^n$, the Euler characteristic is given by an alternating sum of degrees of gradient maps on generic restrictions of the defining polynomial.
• The weight equivariant Euler polynomial of the Milnor fiber $F$ can be computed combinatorially from the intersection lattice of the arrangement when $f$ is $\mathcal{A}'$-generic.