## Publication Date:

2019

## Journal:

Journal of Algebraic Geometry

## Last Updated:

2021-06-02T03:52:54.443+01:00

## Volume:

28

## DOI:

10.1090/jag/737

## page:

405-438

## 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.

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.

## Symplectic id:

401039

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## Submitted to ORA:

Submitted

## Publication Type:

Journal Article