The conjecture on torsion in first homology of Milnor fiber boundaries

Let $\mathcal{A}$ be a central hyperplane arrangement in $\mathbb{C}^{3}$ with $n$ hyperplanes, and let $\partial \overline{F}$ be its Milnor fiber boundary. For $X\in L_{2}(\mathcal{A})$, write $\mathcal{A}_{X}$ for the subarrangement of hyperplanes containing $X$. The projectivized complement is $\mathbb{C}P^{2}\setminus\bigcup_{H\in\mathcal{A}}\overline{H}$.

Torsion conjecture for Milnor fiber boundaries. (1) If

$$(|\mathcal{A}_{X}|-2)(\gcd(|\mathcal{A}_{X}|,n)-1)=0$$

for every $X\in L_{2}(\mathcal{A})$, then the torsion subgroup of $H_{1}(\partial\overline{F};\mathbb{Z})$ is a direct sum of $\mathbb{Z}_{n}$, with the number of summands equal to

$$\chi\left(\mathbb{C}P^{2}\setminus\bigcup_{H\in\mathcal{A}}\overline{H}\right).$$

In particular, for any arrangement with $n$ prime, the torsion subgroup is a direct sum of copies of $\mathbb{Z}_{n}$. (2) If $\mathbb{Z}_{m}$ is contained in the torsion subgroup of $H_{1}(\partial\overline{F};\mathbb{Z})$, then $m$ divides $n$. (3) $H_{1}(\partial\overline{F};\mathbb{Z})$ is torsion free if and only if $\mathcal{A}$ is either a pencil or a near pencil.

The conjecture seeks a combinatorial description of the torsion in the first homology of Milnor fiber boundaries. Its first part gives an explicit formula under a local multiplicity and divisibility condition and includes the prime-$n$ case; the remaining assertions constrain possible torsion orders and characterize torsion-free examples. The problem is open, and the paper presents a theorem for generic arrangements rather than resolving this conjecture in full.