On the reducibility of characteristic varieties (Q2782634)

Let \(V\) be a smooth complex variety with an algebraic Whitney stratification \({\mathcal S}\) and conormal variety \(\Lambda\subset T^*V\). Let \(\Lambda_S=\overline{T_S^*V}\) be the component of \(\Lambda\) lying over \(S\in{\mathcal S}\). If \(\widetilde{\Lambda}\) is the smooth part of \(\Lambda\), set \(\widetilde{\Lambda}_S=\widetilde{\Lambda}\cap\Lambda_S\). If two strata \(S\) and \(T\) are such that \(\dim_{\mathbb C}(\Lambda_S\cap\Lambda_T)=\dim_{\mathbb C}\Lambda-1\), then \(S\) and \(T\) are said to meet microlocally in codimension one. If \({\mathbb P}\) is an \({\mathcal S}\)-constructible perverse sheaf on \(V\), let \({\mathcal M}_S({\mathbb P})\) be the Morse (or vanishing cycles) local system of \({\mathbb P}\) at the stratum \(S\). NEWLINENEWLINENEWLINEThe main result: If \({\mathbb P}\) is an \({\mathcal S}\)-constructible perverse sheaf on \(V\) and \(S\) meets \(\{0\}\) microlocally in codimension one, and the monodromy of \({\mathcal M}_S({\mathbb P})\) around the loop \(\gamma(\theta)=(e^{i\theta},\xi)\) in \(\widetilde{\Lambda}_S\ni(x,\xi)\) is not multiplication by \((-1)^{d-1}\), then \({\mathcal M}_0{\mathbb P}\neq 0\). NEWLINENEWLINENEWLINEThis result gives a unified explanation for the reducible characteristic varieties of Schubert varieties given by Kashiwara and Saito in the case if \(\text{ SL}_n\) and by Boe and Fu for the Lagrangian Grassmannian.