Highly connected non-formal Milnor fibers via polyhedral products

We show that the realization theorem of Fernández de Bobadilla, which identifies the Milnor fiber of a weighted-homogeneous polynomial with the complement of a germ of an analytic set, can be combined with the systematic Massey product constructions of Grbić–Linton for moment-angle complexes $$\mathscr {Z}_K= \mathscr {Z}_K(D^2\!, S^1)$$ to produce weighted-homogeneous polynomials whose Milnor fibers are arbitrarily highly connected and non-formal. The original application of this strategy, due to Fernández de Bobadilla, used the Denham–Suciu classification of lowest-degree triple Massey products and yielded only 2-connected non-formal Milnor fibers. The Grbić–Linton framework, which constructs non-trivial n-fold Massey products in $$H^*(\mathscr {Z}_K;\mathbb {Q})$$ for arbitrary n and in arbitrary cohomological degrees, removes this connectivity restriction entirely. For the explicit families we construct, we also compute the dimension of the singular set and show that the resulting Kato–Matsumoto connectivity bound is sharp.