On motivic vanishing cycles of critical loci

Let $U$ be a smooth scheme over an algebraically closed field $\mathbb K$ of characteristic zero and let $f:U\rightarrow \mathbb A^1$ be a regular function, and write $X=\textrm {Crit}(f)$, as a closed $\mathbb K$-subscheme of $U$. The motivic vanishing cycle $MF_{U,f}^\textrm {mot,\phi }$ is an element of the $\hat \mu$-equivariant motivic Grothendieck ring $\mathcal M^{\hat \mu }_X$, defined by Denef and Loeser, and Looijenga, and used in Kontsevich and Soibelman’s theory of motivic Donaldson–Thomas invariants.

We prove three main results:

(a) $MF_{U,f}^\textrm {mot,\phi }$ depends only on the third-order thickenings $U^{(3)},f^{(3)}$ of $U,f$.

(b) If $V$ is another smooth $\mathbb K$-scheme, $g:V\rightarrow \mathbb A^1$ is regular, $Y=\textrm {Crit}(g)$, and $\Phi :U\rightarrow V$ is an embedding with $f=g\circ \Phi$ and $\Phi \vert _X:X\rightarrow Y$ an isomorphism, then $\Phi \vert _X^*\bigl (MF^\textrm {mot, \phi }_{V,g}\bigr )=MF^\textrm {mot, \phi }_{U,f}\odot \Upsilon (P_\Phi )$ in a certain quotient ring $\overline {\!\!\mathcal M\!} ^{\hat \mu }_X$ of $\mathcal M^{\hat \mu }_X$, where $P_\Phi \rightarrow X$ is a principal $\mathbb Z_2$-bundle associated to $\Phi$ and $\Upsilon :\{$principal $\mathbb Z_2$-bundles on $X\}\rightarrow \overline {\!\!\mathcal M\!} ^{\hat \mu }_X$ a natural morphism.

(c) If $(X,s)$ is an oriented algebraic d-critical locus in the sense of Joyce, there is a natural motive $MF_{X,s}\in \overline {\!\!\mathcal M\!} ^{\hat \mu }_X$, such that if $(X,s)$ is locally modelled on $\textrm {Crit}(f:U\rightarrow \mathbb A^1)$, then $MF_{X,s}$ is locally modelled on $MF_{U,f}^\textrm {mot,\phi }$.

Using results of Pantev, Toën, Vaquié, and Vezzosi, 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, and on intersections $L\cap M$ of oriented Lagrangians $L,M$ in an algebraic symplectic manifold $(S,\omega )$.