On motivic vanishing cycles of critical loci
Authors: Bussi Vittoria , Joyce Dominic , Meinhardt Sven 
Published: March 2019

Let

U
U

be a smooth scheme over an algebraically closed field

K

\mathbb K

of characteristic zero and let

f
:
U
→

A

1

f:U\rightarrow \mathbb A^1

be a regular function, and write

X
=

Crit

(
f
)

X=\textrm {Crit}(f)

, as a closed

K

\mathbb K

-subscheme of

U
U

. The motivic vanishing cycle

M

F

U
,
f

mot,\phi 

MF_{U,f}^\textrm {mot,\phi }

is an element of the

μ
^

\hat \mu

-equivariant motivic Grothendieck ring

M

X

μ
^

\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)

M

F

U
,
f

mot,\phi 

MF_{U,f}^\textrm {mot,\phi }

depends only on the third-order thickenings

U

(
3
)

,

f

(
3
)

U^{(3)},f^{(3)}

of 

U
,
f

U,f

.

(b) If

V
V

is another smooth

K

\mathbb K

-scheme,

g
:
V
→

A

1

g:V\rightarrow \mathbb A^1

is regular,

Y
=

Crit

(
g
)

Y=\textrm {Crit}(g)

, and

Φ
:
U
→
V

\Phi :U\rightarrow V

is an embedding with

f
=
g
∘
Φ

f=g\circ \Phi

and

Φ

|
X

:
X
→
Y

\Phi \vert _X:X\rightarrow Y

an isomorphism, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi vertical-bar Subscript upper X Baseline Superscript asterisk Baseline left-parenthesis upper M upper F Subscript upper V comma g Superscript mot comma reverse-solidus phi Baseline right-parenthesis equals upper M upper F Subscript upper U comma f Superscript mot comma reverse-solidus phi Baseline circled-dot normal upper Upsilon left-parenthesis upper P Subscript normal upper Phi Baseline right-parenthesis">

Φ

|
X
∗

(

M

F

V
,
g

mot, \phi 

)

=
M

F

U
,
f

mot, \phi 

⊙
Υ
(

P
Φ

)

\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

M

¯

X

μ
^

\,\,\overline {\!\!\mathcal M\!}\,^{\hat \mu }_X

of

M

X

μ
^

\mathcal M^{\hat \mu }_X

, where

P
Φ

→
X

P_\Phi \rightarrow X

is a principal

Z

2

\mathbb Z_2

-bundle associated to

Φ
\Phi

and

Υ
:
{

\Upsilon :\{

principal

Z

2

\mathbb Z_2

-bundles on

X
}
→

M

¯

X

μ
^

X\}\rightarrow \,\,\overline {\!\!\mathcal M\!}\,^{\hat \mu }_X

a natural morphism.

(c) If

(
X
,
s
)

(X,s)

is an oriented algebraic d-critical locus in the sense of Joyce, there is a natural motive

M

F

X
,
s

∈

M

¯

X

μ
^

MF_{X,s}\in \,\,\overline {\!\!\mathcal M\!}\,^{\hat \mu }_X

, such that if

(
X
,
s
)

(X,s)

is locally modelled on

Crit

(
f
:
U
→

A

1

)

\textrm {Crit}(f:U\rightarrow \mathbb A^1)

, then

M

F

X
,
s

MF_{X,s}

is locally modelled on 

M

F

U
,
f

mot,\phi 

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
∩
M

L\cap M

of oriented Lagrangians

L
,
M

L,M

in an algebraic symplectic manifold

(
S
,
ω
)

(S,\omega )

.