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- W4301478505 abstract "Let $U$ be a smooth scheme over an algebraically closed field $mathbb K$ of characteristic zero and $f:Uto{mathbb A}^1$ a regular function, and write $X=$Crit$(f)$, as a closed subscheme of $U$. The motivic vanishing cycle $MF_{U,f}^phi$ is an element of the $hatmu$-equivariant motivic Grothendieck ring ${mathcal M}^{hatmu}_X$ defined by Denef and Loeser math.AG/0006050 and Looijenga math.AG/0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) $MF_{U,f}^phi$ depends only on the third-order thickenings $U^{(3)},f^{(3)}$ of $U,f$. (b) If $V$ is another smooth scheme, $g:Vto{mathbb A}^1$ is regular, $Y=$Crit$(g)$, and $Phi:Uto V$ is an embedding with $f=gcircPhi$ and $Phivert_X:Xto Y$ an isomorphism, then $Phivert_X^*(MF_{V,g}^phi)$ equals $MF_{U,f}^phi$ twisted by a motive associated to a principal ${mathbb Z}_2$-bundle defined using $Phi$, where now we work in a quotient ring $bar{mathcal M}^{hatmu}_X$ of ${mathcal M}^{hatmu}_X$. (c) If $(X,s)$ is an oriented algebraic d-critical locus in the sense of Joyce arXiv:1304.4508, there is a natural motive $MF_{X,s} inbar{mathcal M}^{hatmu}_X$, such that if $(X,s)$ is locally modelled on Crit$(f:Uto{mathbb A}^1)$, then $MF_{X,s}$ is locally modelled on $MF_{U,f}^phi$. Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with orientation data, as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090." @default.
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- W4301478505 date "2013-05-28" @default.
- W4301478505 modified "2023-09-29" @default.
- W4301478505 title "On motivic vanishing cycles of critical loci" @default.
- W4301478505 doi "https://doi.org/10.48550/arxiv.1305.6428" @default.
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