On motivic vanishing cycles of critical loci (1305.6428v3)

Abstract: Let $U$ be a smooth scheme over an algebraically closed field $\mathbb K$ of characteristic zero and $f:U\to{\mathbb A}^1$ 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 $\hat\mu$-equivariant motivic Grothendieck ring ${\mathcal M}^{\hat\mu}_X$ 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, $g:V\to{\mathbb A}^1$ is regular, $Y=$Crit$(g)$, and $\Phi:U\to V$ is an embedding with $f=g\circ\Phi$ and $\Phi\vert_X:X\to Y$ an isomorphism, then $\Phi\vert_X^*(MF_{V,g}^\phi)$ 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 $\bar{\mathcal M}^{\hat\mu}_X$ of ${\mathcal M}^{\hat\mu}_X$. (c) If $(X,s)$ is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive $MF{X,s} \in\bar{\mathcal M}^{\hat\mu}_X$, such that if $(X,s)$ is locally modelled on Crit$(f:U\to{\mathbb A}^1)$, 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.