Orlik's integral diagonalization conjecture for weighted homogeneous monodromy

Assume Y=Cn+1Y={\mathbb C}^{n+1} and that ff is weighted homogeneous. Let CC be the matrix of the monodromy operator hh_* on Hn(Xδ,C)H_n(X_\delta,{\mathbb C}) with respect to a basis, let μ\mu be the dimension of this homology group, and let II be the μ×μ\mu\times\mu identity matrix. Orlik's conjecture. The matrix CC can be diagonalized over the integers: there exist unimodular matrices U(t)U(t) and V(t)V(t) with entries in Z[t]{\mathbb Z}[t] such that

U(t)(tIC)V(t)=diag(m1(t),,mμ(t)),U(t)(tI-C)V(t)={\rm diag}(m_1(t),\ldots,m_\mu(t)),

where mi(t)m_i(t) divides mi+1(t)m_{i+1}(t) for i=1,,μ1i=1,\ldots,\mu-1. This would give an integral Smith-type decomposition of the monodromy matrix in the weighted homogeneous case; the source provides no resolution of the conjecture.

Sources & referencesView supporting material

Primary source

Wolfgang Ebeling, “Monodromy”, arXiv:math/0507171 (2005).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.