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• W3105393485 abstract "Let $U$ be a smooth $mathbb C$-scheme, $f:Utomathbb A^1$ 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 $Phi:Uto U$ is an isomorphism with $fcircPhi=f$ and $Phivert_X=$id$_X$. Then $Phi$ induces an isomorphism $Phi_*:PV_{U,f}to PV_{U,f}$. We show that $Phi_*$ is multiplication by det$(dPhivert_X)=1$ or $-1$. (b) $PV_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$, for $X^{(3)}$ the third-order thickening of $X$ in $U$, and $f^{(3)}=fvert_{X^{(3)}}:X^{(3)}tomathbb A^1$. (c) If $U,V$ are smooth $mathbb C$-schemes, $f:Utomathbb A^1$, $g:Vtomathbb A^1$ are regular, $X=$Crit$(f)$, $Y=$Crit$(g)$, and $Phi:Uto V$ is an embedding with $f=gcircPhi$ and $Phivert_X:Xto Y$ an isomorphism, there is a natural isomorphism $Theta_Phi:PV_{U,f}toPhivert_X^*(PV_{V,g})otimes_{mathbb Z_2}P_Phi$, for $P_Phi$ a natural principal $mathbb Z_2$-bundle on $X$. (d) If $(X,s)$ is an oriented d-critical locus in the sense of Joyce arXiv:1304.4508, there is a natural perverse sheaf $P_{X,s}$ on $X$, such that if $(X,s)$ is locally modelled on Crit$(f:Utomathbb A^1)$ then $P_{X,s}$ is locally modelled on $PV_{U,f}$. We also generalize our results to replace $U,X$ by complex analytic spaces, and $PV_{U,f}$ by $mathcal D$-modules, or mixed Hodge modules. We discuss applications of (d) to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to defining a 'Fukaya category' of Lagrangians in a complex symplectic manifold using perverse sheaves. This is the third in a series of papers arXiv:1304.4508, arXiv:1305.6302, arXiv:1305.6428, arXiv:1312.0090, arXiv:1403.2403, arXiv:1404.1329, arXiv:1504.00690." @default.
• W3105393485 created "2020-11-23" @default.
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• W3105393485 date "2015-01-01" @default.
• W3105393485 modified "2023-10-16" @default.
• W3105393485 title "Symmetries and stabilization for sheaves of vanishing cycles" @default.
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• W3105393485 doi "https://doi.org/10.5427/jsing.2015.11e" @default.
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