Symmetries and stabilization for sheaves of vanishing cycles

@article{Brav2012SymmetriesAS,
  title={Symmetries and stabilization for sheaves of vanishing cycles},
  author={Christopher Brav and Vittoria Bussi and Delphine Dupont and Dominic Joyce and Bal{\'a}zs Szendrői},
  journal={arXiv: Algebraic Geometry},
  year={2012}
}
Let $U$ be a smooth $\mathbb C$-scheme, $f:U\to\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\mathbb C$-subscheme of $U$. Then one can define the "perverse sheaf of vanishing cycles" $PV_{U,f}$, a perverse sheaf on $X$. This paper proves four main results: (a) Suppose $\Phi:U\to U$ is an isomorphism with $f\circ\Phi=f$ and $\Phi\vert_X=$id$_X$. Then $\Phi$ induces an isomorphism $\Phi_*:PV_{U,f}\to PV_{U,f}$. We show that $\Phi_*$ is multiplication by det$(d\Phi… 
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