On motivic vanishing cycles of critical loci

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 $M{F}_{U,f}^{\varphi }$ is an element of the $\hat{\mu }$-equivariant motivic Grothendieck ring ${\mathcal{M}}_X^{\hat{\mu }}$ 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) $M{F}_{U,f}^{\varphi }$ 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 $\Phi {|}_X:X\to Y$ an isomorphism, then $\Phi {|}_X^{\ast }(M{F}_{V,g}^{\varphi })$ equals $M{F}_{U,f}^{\varphi }$ "twisted" by a motive associated to a principal ${\mathbb{Z}}_2$-bundle defined using $\Phi$, where now we work in a quotient ring ${\bar{\mathcal{M}}}_X^{\hat{\mu }}$ of ${\mathcal{M}}_X^{\hat{\mu }}$. (c) If $(X,s)$ is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive $M{F}_{X,s}\in {\bar{\mathcal{M}}}_X^{\hat{\mu }}$, such that if $(X,s)$ is locally modelled on Crit$(f:U\to {\mathbb{A}}^1)$, then $M{F}_{X,s}$ is locally modelled on $M{F}_{U,f}^{\varphi }$. 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.

32 pages. (v3) Final version, to appear in the Journal of Algebraic Geometry. arXiv admin note: text overlap with arXiv:1211.3259