Orlik's integral diagonalization conjecture for weighted homogeneous monodromy

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Assume Y=Cn+1Y={\mathbb C}^{n+1} and that ff is weighted homogeneous. Let CC be the matrix of the monodromy operator h∗h_* 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)(tI−C)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,…,μ−1i=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.

References

Primary source

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

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