On motivic vanishing cycles of critical loci


Bussi, V
Joyce, D
Meinhardt, S

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Journal of Algebraic Geometry

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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
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.

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Journal Article