# Quantum Lefschetz without curves

Jeongseok Oh and Richard P. Thomas

###### Abstract.

Given one quasi-smooth derived space cut out of another by a section of a
2-term complex of bundles, we give two formulae for its virtual cycle.

They are modelled on the the $p$-fields construction of Chang-Li and the
Quantum Lefschetz principle, and recover these when applied to moduli spaces
of (stable or quasi-) maps. When the complex is a single bundle we recover
results of Kim-Kresch-Pantev.

## Introduction

The original Quantum Lefschetz principle [Ko, Giv, Kim] applied to curves in a
quintic threefold $Q\subset\mathbb{P}^{4}$ cut out by a quintic equation
$s_{Q}\in H^{0}(\mathcal{O}_{\mathbb{P}^{4}}(5))$. Let $\iota\colon
M_{Q}\hookrightarrow M_{\mathbb{P}^{4}}$ denote the moduli spaces of genus $g$
stable maps to $Q$ and $\mathbb{P}^{4}$ in some fixed degree. When $g=0$ then
$M_{\mathbb{P}^{4}}$ carries a natural bundle $E$ and section $s\in H^{0}(E)$
which, over the point $(f\colon C\to\mathbb{P}^{4})$, have fibre

(0.1) $f^{*}s_{Q}\ \in\
H^{0}\big{(}f^{*}\mathcal{O}_{\mathbb{P}^{4}}(5)\big{)}.$

Clearly (0.1) vanishes if and only if $\operatorname{im}f\subset Q$, so $s$
cuts $M_{Q}$ out of $M_{\mathbb{P}^{4}}$ set-theoretically. In fact this is
true as _Deligne-Mumford stacks with perfect obstruction theory_ : $s^{-1}(0)$
inherits a natural perfect obstruction theory which agrees with the one on
$M_{Q}$. As a result its virtual cycle can be computed in terms of data on
$M_{\mathbb{P}^{4}}$:

(0.2) $\iota_{*}[M_{Q}]^{\operatorname{vir}}\ =\
e(E)\cap[M_{\mathbb{P}^{4}}].$

This aids computation because $\mathbb{P}^{4}$ carries a torus action ($Q$
does not).

When $g\geq 1$ the bundle $E$ is replaced by a _2-term complex of vector
bundles_ $E_{\bullet}$ over $M_{\mathbb{P}^{4}}$, which at the point $(f\colon
C\to\mathbb{P}^{4})$ computes $H^{*}(f^{*}\mathcal{O}_{\mathbb{P}^{4}}(5))$.
Motivated by work of Guffin-Sharpe-Witten [GS], Chang-Li [CL1] moved the
problematic $H^{1}$ term111Note this term in degree 2 is surjected onto by the
obstruction sheaf of $M_{\mathbb{P}^{4}}$, so the virtual tangent bundle of
$M_{Q}$ is still supported in only degrees 0, 1. from degree 2 in the virtual
tangent bundle of $(f^{*}s_{Q})^{-1}(0)$ to degree 0 and dualised, forming a
moduli space $F$ of stable maps to $\mathbb{P}^{4}$ with “$p$-fields”. This is
a cone over $M_{\mathbb{P}^{4}}$ whose fibre over $(f\colon
C\to\mathbb{P}^{4})$ is

(0.3) $H^{0}\big{(}\omega_{C}\otimes
f^{*}\mathcal{O}_{\mathbb{P}^{4}}(-5)\big{)}\ \cong\
H^{1}\big{(}f^{*}\mathcal{O}_{\mathbb{P}^{4}}(5)\big{)}^{*}.$

It comes with a natural perfect obstruction theory on which $f^{*}s_{Q}$
induces a natural _cosection_ [KL1] whose zero locus is $M_{Q}\subset
M_{\mathbb{P}^{4}}\subset F$. Thus by cosection localisation [KL1] we get a
virtual cycle $[F]^{\operatorname{loc}}\in A_{0}(M_{Q})$ localised to $M_{Q}$,
which Chang-Li show recovers $M_{Q}$’s natural virtual cycle,

$[M_{Q}]^{\operatorname{vir}}\ =\ (-1)^{e}\hskip
0.6pt[F]^{\operatorname{loc}}\ \in\ A_{0}(M_{Q}),$

where $e:=\operatorname{rank}E_{\bullet}$. Although they did not give an
analogue of (0.2) (but see the result (0.5) below), we have gained something:
by expressing $[M_{Q}]^{\operatorname{vir}}$ in terms of data on
$M_{\mathbb{P}^{4}}$ we can apply torus localisation. This has lead to great
progress [CGL, CL2, GJR, KL2] in computing higher genus Gromov-Witten
invariants of the quintic.

Others [BN, CG, CJW, CL3, Ke, KO, Lee, Pi1] have generalised this
construction, but always for moduli of (stable or quasi-) maps of curves to a
variety. We assumed this restriction was required to produce the cosection,
but it turns out to exist more generally. The general setup replaces
$M_{Q}\subset M_{P},\,E_{\bullet},\,s,\,F/M_{P}$ by data
$\mathbf{M}\subset\mathbf{P},\,E_{\bullet},\,s,\,\mathbf{F}/\mathbf{P}$ as
follows. We fix

  * •

a quasi-smooth222A quasi-smooth derived structure $\bf P$ on an underlying
scheme or stack $P$ is a slight enhancement of a perfect obstruction theory on
$P$ which always exists in nature and seems to be necessary to set up our
problem correctly. ambient derived Deligne-Mumford stack $\mathbf{P}$ of
(virtual) dimension $p$, whose underlying stack $P:=\pi_{0}({\bf P})$ has the
resolution property (see [Kr, Proposition 5.1] for equivalent conditions),

  * •

an object $E_{\bullet}\in D(\mathrm{coh}\,\mathbf{P})$ of rank $e$, quasi-
isomorphic on $P$ to a 2-term complex of vector bundles $E_{0}\to E_{1}$,

  * •

a section $s\in H^{0}(E_{\bullet})$ inducing a derived structure of dimension
$p-e$ on its zero locus
$\iota\colon\mathbf{M}:=s^{-1}(0)\hookrightarrow\mathbf{P}$.

Let
$\mathbf{F}:=\operatorname{Spec}\operatorname{Sym}^{\bullet}(E_{\bullet}[1])\xrightarrow{\
\pi\ }\mathbf{P}$ be the quasi-smooth total space of the derived dual
$E_{\bullet}^{\vee}[-1]$. The fibre of the underlying stack
$F\subset\mathbf{F}$ over a point $x\in P$ is
$H^{1}\big{(}(E_{\bullet})_{x}\big{)}^{*}$, just as in (0.3). The section
$s\in H^{0}(E_{\bullet})$ is equivalent to a shifted function $\tilde{s}\in
H^{0}\big{(}\mathcal{O}_{\mathbf{F}}[-1]\big{)}$ on $\mathbf{F}$, linear on
the fibres, via

$H^{0}(E_{\bullet})\ \subset\
H^{0}\big{(}\operatorname{Sym}^{\bullet}(E_{\bullet}[1])[-1]\big{)}\ =\
H^{0}\big{(}\pi_{*}\hskip 0.6pt\mathcal{O}_{\mathbf{F}}[-1]\big{)}\ =\
H^{0}\big{(}\mathcal{O}_{\mathbf{F}}[-1]\big{)}.$

Its derivative gives a map $d\hskip
0.6pt\tilde{s}\colon\mathbb{T}_{\mathbf{F}}\to\mathcal{O}_{\mathbf{F}}[-1]$.
Taking $h^{1}$ therefore gives a map from the obstruction sheaf of
$\mathbf{F}$ to its functions, i.e. a cosection [KL1].

######  Theorem.

The cosection $h^{1}(d\tilde{s})\colon
h^{1}(\mathbb{T}_{\mathbf{F}})|_{F}\to\mathcal{O}_{F}$ has image the ideal
sheaf of $M\subseteq F$ if and only if $\mathbf{M}$ is quasi-smooth. In this
case its cosection-localised virtual cycle is

(0.4) $[\mathbf{F}]^{\operatorname{loc}}\ =\ (-1)^{e}\hskip
0.6pt[\mathbf{M}]^{\operatorname{vir}}\ \in\ A_{\hskip
0.6ptp-e}(M,\mathbb{Q})$

Moreover we have the following analogue of (0.2),

(0.5) $\iota_{*}\big{(}e(E_{1})\cap[\mathbf{M}]^{\operatorname{vir}}\big{)}\
=\ e(E_{0})\cap[\mathbf{P}]^{\operatorname{vir}}\ \in\ A_{\hskip 0.6ptp-
e_{0}}(P).$

When $E_{\bullet}=E_{0}$ is a bundle the formula (0.5) is a result of Kim-
Kresch-Pantev [KKP]. In general (0.5) is the best we can do — we cannot expect
a formula for the bare $[\mathbf{M}]^{\operatorname{vir}}$ in terms of
$[\mathbf{P}]^{\operatorname{vir}}$ since the latter will have smaller
dimension than the former when $E_{\bullet}=E_{1}[-1]$.

The first part of the Theorem describes $M$ as the critical locus of
$\tilde{s}$, giving it a derived structure of dimension $2(p-e)$: _twice_ that
of $\mathbf{M}$. In fact this derived structure on $M$ is the $(-2)$-shifted
cotangent bundle $T^{*}[-2]\hskip 0.6pt\mathbf{M}$. In Section 3 we explain
that the main idea behind the equality (0.4) is that $T^{*}[-2]\hskip
0.6pt\mathbf{M}$ admits a virtual cycle [OT1] computed using an auxiliary
maximal isotropic subbundle of a certain bundle with quadratic form. Choosing
one such subbundle gives $[\mathbf{M}]^{\operatorname{vir}}$, using another
leads to $(-1)^{e}\hskip 0.6pt[\mathbf{F}]^{\operatorname{loc}}$.

It should perhaps not be a surprise that $(-2)$-shifted symplectic geometries
[PTVV] — and their associated virtual cycles [BJ, OT1] — should play a role,
given their relationship to cosections [KP, Pi2] and to closed $(-1)$-shifted
1-forms and shifted critical loci. For this and other reasons this paper is in
many ways just the $(-1)$-shift of the paper [JT].

We can apply our results to curve counting to recover the results of [CJW,
Pi1]. Let $M_{P}$ be the moduli space of stable maps of fixed degree and genus
to a smooth projective variety $P$, with universal curve and map

---
$\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{f}$$\textstyle{P}$$\textstyle{M_{P}.\\!\\!}$

Let $(E,s)$ be a bundle and regular section over $P$ with smooth zero locus
$Q\subset P$. Let $(E_{\bullet}\hskip 0.6pt,s)$ on $M_{P}$ be defined by the
composition

$\mathcal{O}_{M_{P}}\xrightarrow{\ \pi^{*}\
}R\pi_{*}\mathcal{O}_{\mathcal{C}}\xrightarrow{\ R\pi_{*}f^{*}s\
}R\pi_{*}f^{*}E\ =:\ E_{\bullet}\,.$

Then $s$ has zero locus $M_{Q}\subset M_{P}$ the stable maps to $Q$ and (0.4)
applies. There is a similar story for quasimaps but with $(E,s)$ defined on
Artin quotient stacks $Q\subset P$. The same construction induces
$(E_{\bullet},s)$ cutting $M_{Q}$ out of the quasi-smooth Deligne-Mumford
stack $M_{P}$, so again (0.4) applies.

### Plan of paper

Section 1 proves (0.5), while Section 2 shows the cosection
$h^{1}(d\tilde{s})$ cuts out $M\subset F$. This leaves the proof of (0.4) to
Section 3 (if the local Kuranishi model can be globalised) and Section 4 (in
general).

By now both “quantum” and “Lefschetz” are both sufficiently far from our
results that they should be thought of merely as motivation; “virtual Euler”
might be more appropriate.

We denote the derived dual $R\curly H\\!om(E,\mathcal{O})$ of an object $E$ by
$E^{\vee}$, but use $E^{*}$ in the special case of vector spaces and bundles.

We use $\mathbb{Q}$ coefficients for our Chow groups throughout, but it should
be noted that when $P$ is a scheme all results hold with $\mathbb{Z}$
coefficients. This uses the existence of the global maximal isotropic
subbundle $\Lambda$ (4.16) in Section 4, which ensures the results of [OT1]
hold with $\mathbb{Z}$ coefficients; see for instance [OT1, Equation (34)].
(In general [OT1] works with $\mathbb{Z}[\frac{1}{2}]$-coefficients — results
are proved on a certain bundle over $M$ on which a $\Lambda$ exists; inverting
2 then allows the descent back down to $M$.)

Acknowledgements. This paper is dedicated to Bumsig Kim, whose influence is
all over this topic.

We thank Young-Hoon Kiem, Hyeonjun Park and Renata Picciotto for discussions
about cosections, $p$-fields and their closely related preprints [KP, Pi2],
which give a generalisation of our result to the non-quasi-smooth case.

Thanks also to Felix Janda and Bhamidi Sreedhar for useful conversations.

Both authors were supported by funding from a Royal Society research
professorship.

##  1\. Euler classes

In this section we show (0.5) follows easily from work of Kim-Kresch-Pantev.
Let $s_{0}$ denote the projection of $s$ under $E_{\bullet}\to E_{0}$. This
gives $M$ a different quasi-smooth structure333In local Kuranishi models for
these structures, like in (2.2) below, the cdgas for $\mathbf{M}$ and
$\mathbf{M}^{\prime}$ differ only in degrees $-2$ (by an $E_{1}^{*}$ term) and
lower, so $\pi_{0}(\mathbf{M}^{\prime})=\pi_{0}(\mathbf{M})=M$.
$\mathbf{M}^{\prime}$, cut out of $\mathbf{P}$ by $s_{0}\in H^{0}(E_{0})$. By
[KKP] its virtual cycle pushes forwards to the right hand side of (0.5),

$\iota_{*}[\mathbf{M}^{\prime}]^{\operatorname{vir}}\ =\
e(E_{0})\cap[\mathbf{P}]^{\operatorname{vir}}.$

Let
$\mathbf{E}:=\operatorname{Spec}\operatorname{Sym}^{\bullet}E_{\bullet}^{\vee}$
denote the total space of $E_{\bullet}$ with zero section $0_{\hskip
0.6pt\mathbf{E}}$ and section $s$, so that $\mathbf{M}$ is defined by the
derived fibre product

$\textstyle{\mathbf{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{\mathbf{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0_{\hskip
0.6pt\mathbf{E}}}$$\textstyle{\mathbf{E}.\\!\\!}$

This gives the central horizontal exact triangle of the following commutative
diagram of tangent complexes,

(1.1)
---
$\textstyle{\mathbb{T}_{\mathbf{P}}\big{|}_{\mathbf{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,0)}$$\scriptstyle{0_{\hskip
0.6pt\mathbf{E}*}}$$\textstyle{\mathbb{T}_{\mathbf{E}}\big{|}_{\mathbf{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{\mathbf{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{T}_{\mathbf{P}}\oplus\mathbb{T}_{\mathbf{P}})\big{|}_{\mathbf{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0_{\hskip
0.6pt\mathbf{E}*},s_{*})}$$\scriptstyle{(0,1)}$$\textstyle{\mathbb{T}_{\mathbf{E}}\big{|}_{\mathbf{M}}}$$\textstyle{\mathbb{T}_{\mathbf{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{\mathbf{P}}\big{|}_{\mathbf{M}}.}$

The zero section $0_{\mathbf{E}}\colon\mathbf{P}\hookrightarrow\mathbf{E}$ and
the projection $\mathbf{E}\to\mathbf{P}$ together define a splitting of the
tangent complex of $\mathbf{E}$ restricted to $\mathbf{P}$,

$\mathbb{T}_{\mathbf{E}}|_{\mathbf{P}}\ \cong\
\mathbb{T}_{\mathbf{P}}\,\oplus\,\mathbb{T}_{\mathbf{E}/\mathbf{P}}|_{\mathbf{P}}\
\cong\ \mathbb{T}_{\mathbf{P}}\,\oplus\,E_{\bullet}.$

So, up to shifts, the cone on the top row of (1.1) is
$E_{\bullet}|_{\mathbf{M}}$, while the bottom row is
$\mathbb{T}_{\mathbf{M}/\mathbf{P}}$. Since the central row is exact the
upshot is that $\mathbb{T}_{\mathbf{M}/\mathbf{P}}\cong
E_{\bullet}|_{\mathbf{M}}[-1]$ sat in the exact triangle

$\mathbb{T}_{\mathbf{M}}\longrightarrow\mathbb{T}_{\mathbf{P}}|_{\mathbf{M}}\longrightarrow\mathbb{T}_{\mathbf{M}/\mathbf{P}}[1]\,=\,E_{\bullet}.$

Restricting this to $M$ gives the top row of the following diagram of exact
triangles; repeating the working with $(E_{\bullet},s)$ replaced by
$(E_{0},s_{0})$ gives the second row,

$\textstyle{\mathbb{T}_{\mathbf{M}}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{\mathbf{P}}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\bullet}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{\mathbf{M}^{\prime}}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{\mathbf{P}}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{0}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\big{|}_{M}[-1]}$$\textstyle{E_{1}\big{|}_{M}\,.\\!\\!\\!}$

Picking a global locally free resolution $A\to K^{\prime}$ for
$T_{\mathbf{M}^{\prime}}|_{M}$, the composition
$K^{\prime}\to\mathbb{T}_{\mathbf{M}^{\prime}}|_{M}[1]\to E_{1}|_{M}$ is onto,
defining an exact sequence

$0\longrightarrow K\longrightarrow K^{\prime}\longrightarrow
E_{1}|_{M}\longrightarrow 0$

and a resolution $A\to K$ for $\mathbb{T}_{\mathbf{M}}|_{M}$. Dualising,
$\\{K^{*}\to A^{*}\\}=\mathbb{L}_{\mathbf{M}}|_{M}\to\mathbb{L}_{M}$ is a
perfect obstruction theory for $M$ and the construction of Behrend-Fantechi
[BF] gives a cone $C\subset K$ such that
$[\mathbf{M}]^{\operatorname{vir}}=0^{\hskip 0.6pt!}_{K}[C]$. Then

$[\mathbf{M}^{\prime}]^{\operatorname{vir}}\ =\ 0^{\hskip
0.6pt!}_{K^{\prime}}[C]\ =\ e(E_{1}|_{M})\cap 0^{\hskip 0.6pt!}_{K}[C]\ =\
e(E_{1})\cap[\mathbf{M}]^{\operatorname{vir}},$

the second equality by [Fu, Theorem 6.3]. This is the left hand side of (0.5).

##  2\. Cosection

Throughout it will be convenient to extend our $\,\widetilde{}\,$ notation:
given any object $G\in D(\mathrm{coh}\,\mathbf{P})$, a section $\varphi\in
H^{0}(\mathbf{P},\,G\otimes E_{\bullet})$ can be thought of as a shifted
section $\tilde{\varphi}\in H^{0}(\mathbf{F},\,\pi^{*}G[-1])$, linear on the
fibres of $\pi$, via

(2.1) $\displaystyle\varphi$ $\displaystyle\in$ $\displaystyle H^{0}(G\otimes
E_{\bullet})\ \subset\
H^{0}\big{(}G\otimes\operatorname{Sym}^{\bullet}(E_{\bullet}[1])[-1]\big{)}$
$\displaystyle=$ $\displaystyle H^{0}\big{(}G\otimes\pi_{*}\hskip
0.6pt\mathcal{O}_{\mathbf{F}}[-1]\big{)}\ =\ H^{0}\big{(}\pi^{*}G[-1]\big{)}\
\ni\ \tilde{\varphi}.$

In this Section we prove that $h^{1}(d\tilde{s})$ cuts out $M\subseteq F$ if
$\mathbf{M}$ is quasi-smooth. It is enough to work locally, where we can put
everything in a standard model.

$\bullet$ The local model for a quasi-smooth $\mathbf{P}$ is a _Kuranishi
chart_ $(A,B,t)$: a smooth ambient space $A$ over which we have a section $t$
of a bundle $B$ cutting out $\mathbf{P}$ in the sense that its structure sheaf
is (quasi-isomorphic as a cdga to) the Koszul complex

(2.2) $\mathcal{O}_{\mathbf{P}}\ \cong\ (\Lambda^{\bullet}B^{*},t)\ :=\
\big{\\{}\cdots\longrightarrow\Lambda^{2}B^{*}\xrightarrow{\ t\
}B^{*}\xrightarrow{\ t\ }\mathcal{O}_{A}\big{\\}}.$

The cotangent complex of $\mathbf{P}$ is most easily described when $A$ is
sufficiently small that $B$ admits a connection $D$. Then

(2.3) $\mathbb{L}_{\mathbf{P}}\ =\ \big{(}B^{*}\xrightarrow{\ \cdot Dt\
}\Omega_{A}\big{)}\otimes_{\mathcal{O}_{A}}(\Lambda^{\bullet}B^{*},t)$

is a differential graded module over the dga $(\Lambda^{\bullet}B^{*},t)$ in
the obvious way. The exterior derivative acts on degree 0 functions by
$\mathcal{O}_{A}\ni f\mapsto df\otimes 1\in\Omega_{A}\otimes\mathcal{O}_{A}$
and on degree $(-1)$ functions by

(2.4) $B^{*}\,\in\,f\,\ \longmapsto\ \,(f\otimes 1)\,\oplus\,Df\ \in\
(B^{*}\otimes\mathcal{O}_{A})\,\oplus\,(\Omega_{A}\otimes B^{*}).$

This can be checked to intertwine the differentials on the dga
$(\Lambda^{\bullet}B^{*},t)$ and the dgm (2.3), and extends to degree
$\leq(-2)$ functions by the Leibniz rule.

$\bullet$ The local model for $E_{\bullet}$ is a complex of bundles $d\colon
E_{0}\to E_{1}$ over $A$ tensored over $\mathcal{O}_{A}$ with (2.2) — i.e. the
total complex $\Lambda^{\bullet}B^{*}\otimes\\{E_{0}\to E_{1}\\}$ with
differential $t\otimes 1-(-1)^{i}\otimes d$ on $\Lambda^{i}B^{*}\otimes
E_{0}$. In degrees 0 and 1 this is

$\cdots\longrightarrow(B^{*}\otimes E_{1})\,\oplus\,E_{0}\xrightarrow{\,t-d\
}E_{1},$

so $s\in H^{0}(E_{\bullet})$ is represented by
$(s_{1},s_{0})\in\Gamma\big{(}(B^{*}\otimes E_{1})\oplus E_{0}\big{)}$ such
that $(t-d)(s_{1},s_{0})=0$. Thus $s_{1}(t)=d\circ s_{0}$ and $s$ is
represented by a commutative diagram of $\mathcal{O}_{A}$-modules

(2.5)
$\textstyle{\mathcal{O}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{0}}$$\scriptstyle{t}$$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{1}}$$\textstyle{E_{1}.\\!\\!}$

$\bullet$ The local model for $\mathbf{F}$ is inside
$\operatorname{tot}_{\\!A}(E_{1}^{*})$, cut out by the section

$r\ :=\ \big{(}p^{*}t,\,-\widetilde{d^{*}}\big{)}\ \in\
\Gamma\big{(}p^{*}B\oplus p^{*}E^{*}_{0}\big{)}.$

Here $p$ is the projection $\operatorname{tot}_{\\!A}(E_{1}^{*})\to A$ on
which we are using the $\,\widetilde{}\,$ notation of (2.1). Thus
$\mathcal{O}_{\mathbf{F}}$ is the associated Koszul cdga
$\big{(}\Lambda^{\bullet}(p^{*}B^{*}\oplus p^{*}E_{0}),\,r\big{)}$ on
$\operatorname{tot}_{\\!A}(E_{1}^{*})$. In its degree $(-1)$ piece lies the
$(-1)$-shifted function $\tilde{s}$,

$\tilde{s}\ =\ \tilde{s}_{1}+p^{*}s_{0}\ \in\ \Gamma\big{(}p^{*}B^{*}\oplus
p^{*}E_{0}\big{)}.$

By (2.4) we can read off its exterior derivative $d\tilde{s}$ in

$\mathbb{L}_{\mathbf{F}}\ =\ \Big{(}p^{*}B^{*}\oplus
p^{*}E_{0}\longrightarrow\Omega_{\mathrm{tot}_{\\!A}(E_{1}^{*})}\Big{)}\otimes\Lambda^{\bullet}(p^{*}B^{*}\oplus
p^{*}E_{0}).$

It lies in the degree $(-1)$ part

$(p^{*}B^{*}\otimes\mathcal{O})\ \oplus\ (p^{*}E_{0}\otimes\mathcal{O})\
\oplus\ (\Omega\otimes p^{*}B^{*})\ \oplus\ (\Omega\otimes p^{*}E_{0}),$

with respect to which it is

$\big{(}\tilde{s}_{1}\otimes 1,\ p^{*}s_{0}\otimes 1,\ \widetilde{Ds_{1}},\
p^{*}Ds_{0}\big{)}.$

Restricting to $F\subset\mathbf{F}$ kills the third and fourth terms (since
they involve degree $(-1)$ functions). So we are left with showing that

(2.6) $(\tilde{s}_{1},p^{*}s_{0})\,\colon\,p^{*}(B\oplus
E^{*}_{0})\longrightarrow\mathcal{O}_{F}$

has image the ideal sheaf of $M$. But $s_{0}$ cuts out $M$ from $P$ so
$p^{*}(s_{0})$ cuts out $F_{M}:=F\times_{P}M$ and what remains to show is that
$\tilde{s}_{1}\colon p^{*}B|_{F_{M}}\to\mathcal{O}_{F_{M}}$ generates the
ideal sheaf of $M\subset F_{M}$. For this we consider the diagram

$\textstyle{p^{*}(B\oplus
E_{0})\big{|}_{F_{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{*}(s_{1},-d)}$--------$\scriptstyle{(1,0)}$$\textstyle{p^{*}B\big{|}_{F_{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{*}(s_{1})}$$\scriptstyle{\tilde{s}_{1}}$$\textstyle{p^{*}E_{1}\big{|}_{F_{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau|_{F_{M}}}$$\textstyle{\mathcal{O}_{F_{M}},}$

where $\tau\in H^{0}(p^{*}E_{1}^{*})$ is the tautological section of
$E_{1}^{*}$ on $\operatorname{tot}_{\\!A}(E_{1}^{*})$. Since $\tau\circ
p^{*}(s_{1})=\tilde{s}_{1}$ and $\tau\circ p^{*}(d\hskip 0.6pt)=\tilde{d}$ —
and the latter vanishes on $F_{M}$ because $\tilde{d}^{*}\colon
p^{*}E_{1}^{*}\to p^{*}E_{0}^{*}$ vanishes on $F$ — it follows that the
diagram commutes. Most importantly, the image of $\tau\colon
p^{*}E_{1}\to\mathcal{O}_{\operatorname{tot}_{\\!A}(E_{1}^{*})}$ is the ideal
of the zero section $A\subset\operatorname{tot}_{\\!A}(E_{1}^{*})$. Now
$\mathbf{M}$ is quasi-smooth if and only if

$h^{2}(\mathbb{T}_{\mathbf{M}})\ =\ 0\ =\
\operatorname{coker}\big{[}(s_{1},-d\hskip 0.6pt)\colon(B\oplus
E_{0})\big{|}_{M}\to E_{1}\big{|}_{M}\big{]};$

see (3.1) below, for instance. So in this case the lower curved arrow is onto
and its composition with $\tau|_{F_{M}}$ generates the ideal of the zero
section $M\subset F|_{M}$. Thus so does the upper curved arrow, as required.

##  3\. Idea of the proof

We explain how (0.4) works in the special case that _the local model ( 2.5)
holds globally._ The main idea is to consider a _third_ derived structure on
$M$, different from both $\mathbf{M}$ and $\mathbf{M}^{\prime}$, namely the
$(-2)$-shifted cotangent bundle $T^{*}[-2]\hskip 0.6pt\mathbf{M}$. This has a
virtual cycle constructed in [OT1] using a choice of maximal isotropic
subbundle of a certain orthogonal bundle. Using one choice will recover
$[\mathbf{M}]^{\operatorname{vir}}$, using another naturally gives
$(-1)^{e}\hskip 0.6pt[\mathbf{F}]^{\operatorname{loc}}$.

From the model (2.5), $M\subset A$ is cut out by the section $(s_{0},t)$ of
the complex $E_{0}\oplus B\to E_{1}$ with differential $(-d,s_{1})$. This
endows it with a derived structure $\mathbf{M}$ with structure sheaf the
Koszul complex

$\mathcal{O}_{\mathbf{M}}\ \cong\
\Big{(}\operatorname{Sym}^{\bullet}\\!\big{\\{}E_{1}^{*}\xrightarrow{}E_{0}^{*}\oplus
B^{*}\big{\\}},(s_{0},t)\Big{)}.$

In particular its tangent complex $\mathbb{T}_{\mathbf{M}}|_{M}$ is

(3.1) $T_{A}\big{|}_{M}\xrightarrow{D(t,s_{0})}B\big{|}_{M}\oplus
E_{0}\big{|}_{M}\xrightarrow{(s_{1},-d)}E_{1}\big{|}_{M}\,$

so $\mathbf{M}$ is quasi-smooth if and only if $(s_{1},-d)\colon(B\oplus
E_{0})|_{M}\to E_{1}|_{M}$ is onto. In this case we set $K$ to be its kernel,
so that

$\mathbb{T}_{\mathbf{M}}\big{|}_{M}\ =\
\big{\\{}T_{A}\big{|}_{M}\xrightarrow{D(t,s_{0})}K\big{\\}}.$

Dualising induces a perfect obstruction theory
$\mathbb{L}_{\mathbf{M}}|_{M}\to\mathbb{L}_{M}$, yielding a Behrend-Fantechi
virtual cycle $[\mathbf{M}]^{\operatorname{vir}}$.

The shifted cotangent bundle $T^{*}[-2]\hskip 0.6pt\mathbf{M}$ has the same
underlying stack $M$ but a different derived structure, with tangent complex

(3.2) $\mathbb{T}_{T^{*}[-2]\hskip 0.6pt\mathbf{M}}\big{|}_{M}\ =\
\big{\\{}T_{A}\big{|}_{M}\xrightarrow{D(t,s_{0})\,\oplus\,0\,}K\oplus
K^{*}\xrightarrow{0\,\oplus\,D(t,s_{0})^{*}}\Omega_{A}\big{|}_{M}\big{\\}}.$

Since $T^{*}[-2]\hskip 0.6pt\mathbf{M}$ is $(-2)$-shifted symplectic it also
admits a virtual cycle [OT1]. This depends on a choice of orientation (in the
sense of [OT1, Section 2]) on the orthogonal bundle $K\oplus K^{*}$ — making
it an $SO(2k,\mathbb{C})$ bundle — and the construction involves picking a
maximal isotropic subbundle (though the final result is independent of it).

There is a canonical orientation $o_{K}$ on $K\oplus K^{*}$ which makes
$K\subset K\oplus K^{*}$ a _positive_ maximal isotropic subbundle [OT1,
Equation (18)]. Then picking $K^{*}\subset K\oplus K^{*}$ as our maximal
isotropic subbundle, $T^{*}[-2]\hskip 0.6pt\mathbf{M}$’s virtual cycle is
$[\mathbf{M}]^{\operatorname{vir}}$ by [OT1, Section 8].

Applying the construction of [OT1] to a different maximal isotropic, however,
will lead naturally to the space $\mathbf{F}$. We begin by replacing (3.2) by
the quasi-isomorphic complex

(3.3) $(T_{A}\oplus E_{1}^{*})\big{|}_{M}\longrightarrow(B\oplus E_{0}\oplus
E_{0}^{*}\oplus B^{*})\big{|}_{M}\longrightarrow(\Omega_{A}\oplus
E_{1})\big{|}_{M}.$

Here the first arrow is the direct sum of $D(t,s_{0})\colon
T_{A}|_{M}\to(B\oplus E_{0})|_{M}$ and $(-d^{*},s_{1}^{*})\colon
E_{1}^{*}|_{M}\to(E_{0}^{*}\oplus B^{*})|_{M}$, and the second arrow is its
dual. We claim (3.3) is the tangent bundle of the $(-2)$-shifted symplectic
derived Deligne-Mumford stack cut out of
$p\colon\operatorname{tot}_{\\!A}(E_{1}^{*})\to A$ by the _isotropic_ section

(3.4) $\sigma\ :=\
\big{(}p^{*}(s_{0}),\,p^{*}(t),\,-\tilde{d}^{*},\,\tilde{s}_{1}^{*}\big{)}\
\text{ of }\ p^{*}(B\oplus E_{0}\oplus E_{0}^{*}\oplus B^{*}).$

Here we use the $\,\widetilde{}\,$ notation of (2.1) and the natural quadratic
form $q$ on $B\oplus E_{0}\oplus E_{0}^{*}\oplus B^{*}$ — pairing $B\oplus
E_{0}$ with its dual — so the commutativity of (2.5) makes $\sigma$ isotropic.
Thus we get a “Darboux chart”

(3.5)
---
$\textstyle{\big{(}p^{*}(B\oplus E_{0}\oplus E_{0}^{*}\oplus
B^{*}),\,p^{*}q\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hskip
28.45274ptp^{*}q\hskip 0.6pt(\sigma,\sigma)=0,}$$\textstyle{M\ =\
\sigma^{-1}(0)\
\subset}$$\textstyle{\mathrm{tot}_{\\!A}(E_{1}^{*}),\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$

such that the two arrows of (3.3) are $D\sigma$ on $M$, which proves our
claim.

The key observation is the following. Let $\Lambda$ denote the maximal
isotropic subbundle $p^{*}(B^{*}\oplus E_{0})$ and split (3.4) as
$\sigma=(\sigma_{1},\sigma_{2})\in H^{0}(\Lambda\oplus\Lambda^{*})$. Then

(3.6) $\sigma_{2}\ =\ \big{(}p^{*}(t),-\tilde{d}^{*}\big{)}\ \in\
H^{0}(\Lambda^{*})\ \text{ cuts out }\mathbf{F}\text{ from
}\operatorname{tot}_{A}(E_{1}^{*}).$

Therefore using $\Lambda$ to define the virtual cycle, the construction of
[OT1, Section 3.2] gives the following (but see Remark 3.9 below). The virtual
cycle of $T^{*}[-2]\hskip 0.6pt\mathbf{M}$ is made by taking the intersection
of

$C_{F/\mathrm{tot}_{\\!A}(E_{1}^{*})}\ \subset\ \Lambda^{*}|_{F}\ \text{ with
the zero section }\ 0_{\Lambda^{*}|_{F}}\,,$

cosection localised by $\sigma_{1}|_{F}\hskip 0.6pt$:

(3.7)
$\pm\,0_{\Lambda^{*}|_{F}}^{\,!,\,\operatorname{loc}}\big{[}C_{F/\mathrm{tot}_{\\!A}(E_{1}^{*})}\big{]}\
\in\ A_{\hskip 0.6ptp-e}(M).$

Here we think of $\sigma_{1}|_{F}$ as a function on $\Lambda^{*}|_{F}$ (linear
on the fibres) which vanishes identically on
$C_{F/\mathrm{tot}_{\\!A}(E_{1}^{*})}$ by [OT1, Lemma 3.1]. Thus the cosection
localisation of [KL1] applies, localising the intersection to the zeros of
$\sigma_{1}|_{F}$ on the zero locus of $\sigma_{2}$, i.e. to the zero locus
$M$ of $\sigma$ as claimed.

We need to describe the sign $\pm$ (3.7), written in [OT1, Section 3.2] as
$(-1)^{|\Lambda|+\operatorname{rank}\Lambda}$. Recall we gave $K\oplus K^{*}$
the orientation $o_{K}$ for which $K$ is a positive maximal isotropic. Under
the passage from (3.2) to (3.3) this corresponds to giving $p^{*}(B\oplus
E_{0}\oplus E_{0}^{*}\oplus B^{*})$ the orientation $o_{K\oplus
p^{*}E_{1}^{*}}=o_{K}\otimes o_{p^{*}E_{1}^{*}}$ by [OT1, Equation (65)].
Writing this as $(-1)^{|\Lambda|}o_{\Lambda}$ defines $(-1)^{|\Lambda|}$.

Working locally we may split the exact sequence $0\to K\to p^{*}(B\oplus
E_{0})\to p^{*}E_{1}\to 0$. Then suppressing some $p^{*}$s and setting
$b=\operatorname{rank}B$, etc,

(3.8) $\displaystyle o_{K}\otimes o_{E_{1}^{*}}$ $\displaystyle=$
$\displaystyle(-1)^{e_{1}}o_{K}\otimes o_{E_{1}}\ =\ (-1)^{e_{1}}o_{K\oplus
E_{1}}\ =\ (-1)^{e_{1}}o_{B\oplus E_{0}}$ $\displaystyle=$
$\displaystyle(-1)^{e_{1}}o_{B}\otimes
o_{E_{0}}=(-1)^{e_{1}+b}o_{B^{*}}\otimes o_{E_{0}}=(-1)^{e_{1}+b}o_{\Lambda}.$

Thus our sign $\pm$ is
$(-1)^{|\Lambda|+\operatorname{rank}\Lambda}=(-1)^{e_{1}+b+b+e_{0}}=(-1)^{e}$
as required.

Finally, by (3.4) we see that
$\sigma_{1}|_{F}=(p^{*}(s_{0}),\,\tilde{s}_{1}^{*})\colon p^{*}(B\oplus
E_{0}^{*})|_{F}\to\mathcal{O}_{F}$, which is the cosection $h^{1}(d\tilde{s})$
as calculated in (2.6). Thus (3.7) is precisely $(-1)^{e}\hskip
0.6pt[\mathbf{F}]^{\operatorname{loc}}$, as required.

######  Remark 3.9.

More precisely, (3.7) is in fact
$(-1)^{|\Lambda|}\sqrt{e}\,(\Lambda\oplus\Lambda^{*},\sigma,\Lambda)$ [OT1,
Section 3.2] — the $\sigma$-localised square root Euler class of
$\Lambda\oplus\Lambda^{*}$. Its construction uses the family of isotropic
graphs $\Gamma_{(\sigma_{1},\,z^{-1}\sigma_{2})}$ in
$\Lambda\oplus\Lambda^{*}$ to interpolate between $\Gamma_{\sigma}$ (at $z=1$)
and its degeneration $C_{F/\mathrm{tot}_{\\!A}(E_{1}^{*})}$ (at $z=0$) — see
the proof of [OT1, Lemma 3.1]. In contrast, the virtual cycle is defined in
[OT1, Section 4.2] by replacing the graph by
$C_{M/\mathrm{tot}_{\\!A}(E_{1}^{*})}$. But this is also a degeneration of
$\Gamma_{\sigma}$ through the family of isotropic graphs
$\Gamma_{\\!z^{-1}\sigma}$ — see [OT1, Equation (71)] — so the result is the
same.

We can combine these two families444The two families lie over
$(w=1)\subset\mathbb{C}^{2}$ and $(z=1)\subset\mathbb{C}^{2}$ respectively.
References for this section are [BCM, KKP, Ma]. over $\mathbb{C}$ into a
single family over $\mathbb{C}^{2}$ by considering the (closure of the)
isotropic graph

$\overline{\Gamma}\ :=\
\overline{\Gamma_{(w^{-1}\sigma_{1},\,(wz)^{-1}\sigma_{2})}}\ \subset\
\mathrm{tot}_{\,\mathrm{tot}_{\\!A}(E_{1}^{*})\times\mathbb{C}^{2}}\big{(}\Lambda\oplus\Lambda^{*}\big{)}.$

Here and below $w,z$ are the coordinates pulled back from $\mathbb{C}^{2}$ and
we suppress some pullback maps on $\Lambda\oplus\Lambda^{*}$. Denote the
inclusion of the point $(w,z)$ into $\mathbb{C}^{2}$ by $i_{w,z}$. Then
factoring $i_{1,0}$ and $i_{0,0}$ through the inclusion of $(z=0)$ gives a
rational equivalence

$i_{1,0}^{\,!}\,\overline{\Gamma}\ \sim\ i_{0,0}^{\,!}\,\overline{\Gamma}\
\text{ inside }\ \overline{\Gamma}\times_{\mathbb{C}^{2}}(z=0)\ \subset\
\mathrm{tot}_{F\times(z=0)}\big{(}\Lambda\oplus\Lambda^{*}\big{)}.$

Similarly factoring $i_{0,1}$ and $i_{0,0}$ through the inclusion of $(w=0)$
gives a rational equivalence

(3.10) $i_{0,1}^{\,!}\,\overline{\Gamma}\ \sim\
i_{0,0}^{\,!}\,\overline{\Gamma}\ \text{ inside }\
\overline{\Gamma}\times_{\mathbb{C}^{2}}(w=0)\ \subset\
\mathrm{tot}_{F\times(w=0)}\big{(}\Lambda\oplus\Lambda^{*}\big{)}.$

But $i_{1,0}^{\,!}\,\overline{\Gamma}=C_{F/\mathrm{tot}_{\\!A}(E_{1}^{*})}$
because $\overline{\Gamma}|_{(w=1)}$ is the (flat!) deformation of
tot${}_{\\!A}(E_{1}^{*})$ to the normal cone
$C_{F/\mathrm{tot}_{\\!A}(E_{1}^{*})}$. Similarly
$i_{0,1}^{\,!}\,\overline{\Gamma}\ =\ C_{M/\mathrm{tot}_{\\!A}(E_{1}^{*})}$
because $\overline{\Gamma}|_{(z=1)}$ is the deformation of
tot${}_{\\!A}(E_{1}^{*})$ to the normal cone
$C_{M/\mathrm{tot}_{\\!A}(E_{1}^{*})}$. The upshot is an _isotropic_ rational
equivalence

(3.11) $C_{F/\mathrm{tot}_{\\!A}(E_{1}^{*})}\ \sim\
C_{M/\mathrm{tot}_{\\!A}(E_{1}^{*})}\ \text{ inside }\
\mathrm{tot}_{F\times(wz=0)}\big{(}\Lambda\oplus\Lambda^{*}\big{)}.$

This will prove useful later because it replaces the ambient space
$\mathrm{tot}_{\\!A}(E_{1}^{*})$ (which only exists locally in general) with
data over $F$ (which will globalise). We note that the embedding of the base
$F$ of $C_{F}$ into $\Lambda\oplus\Lambda^{*}$ factors through the first
factor as $\sigma_{1}=(\pi^{*}s_{0},\,\tilde{s}_{1}^{*})=h^{1}(d\hskip
0.6pt\tilde{s})$.

##  4\. Proof of main result

_Throughout this section we always restrict to the underlying scheme or stack
$Y=\pi_{0}({\bf Y})$ of whichever derived space $\bf Y$ we are working on._ We
usually omit the restriction map for brevity.

Since all of our stacks have the resolution property we may always resolve
(complexes of) sheaves using complexes of very negative locally free sheaves.
So to prove (0.4) in the general case we begin by finding global resolutions
of the objects
$\mathbb{T}_{\mathbf{P}},\,\mathbb{T}_{\mathbf{F}},\,\mathbb{T}_{\mathbf{M}},\,E^{\bullet}$
and the maps between them. We will choose them to be reminiscent of the
(derivative of) the local model (2.5).555The
$A,B,t^{\prime},\mathcal{A},E_{0},E_{1},s_{0},s_{1}$ of this Section play the
roles of the $T_{A},B,dt,T_{\operatorname{tot}_{\\!A}(E_{1}^{*})}$,
$E_{0},E_{1},s_{0},s_{1}$ (all restricted to $P,\,F$ or $M$) of Sections 2 and
3.

First pick a 2-term locally free resolution $E_{0}\xrightarrow{\ d\ }E_{1}$ of
$E_{\bullet}$. Now we may pick a 2-term locally free resolution
$t^{\prime}\colon A\to B$ of $\mathbb{T}_{\mathbf{P}}$ with $B$ sufficiently
negative that the functor $\operatorname{Ext}^{>0}(B,\ \cdot\ )$ is zero on
$E_{0}^{*},\,E_{1}^{*}$ and $\iota_{*}(E_{0}^{*}|_{M}\big{)}$.

### Representative for $\mathbb{T}_{\mathbf{F}}$

The exact triangle
$\mathbb{T}_{\mathbf{F}}\to\pi^{*}\mathbb{T}_{\mathbf{P}}\to\mathbb{T}_{\mathbf{F}/\mathbf{P}}[1]=\pi^{*}E_{\bullet}^{\vee}$
defines an element of the uppermost group in the diagram

$\textstyle{\operatorname{Hom}(\pi^{*}\mathbb{T}_{\mathbf{P}},\pi^{*}E_{\bullet}^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ext}^{1}(\pi^{*}A,\pi^{*}E_{1}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Ext}^{1}(\pi^{*}\mathbb{T}_{\mathbf{P}},\pi^{*}E_{1}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ext}^{1}(\pi^{*}A,\pi^{*}E_{0}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Ext}^{1}(\pi^{*}\mathbb{T}_{\mathbf{P}},\pi^{*}E_{0}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.\\!\\!}$

Here the horizontal arrows come from the triangle $\mathbb{T}_{\mathbf{P}}\to
A\xrightarrow{\ t^{\prime}\ }B$ and the $\operatorname{Ext}^{>0}(B,\ \cdot\ )$
vanishing. Thus we get an extension in
$\operatorname{Ext}^{1}(\pi^{*}A,\pi^{*}E_{1}^{*})$ which maps to zero in
$\operatorname{Ext}^{1}(\pi^{*}A,\pi^{*}E_{0}^{*})$, inducing a commutative
diagram

$\textstyle{\pi^{*}E_{1}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\pi^{*}d^{*}}$$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{*}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{*}E_{0}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,1)}$$\textstyle{\pi^{*}(A\oplus
E_{0}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,0)}$$\textstyle{\pi^{*}A.\\!}$

Composing with $\pi^{*}t^{\prime}\colon\pi^{*}A\to\pi^{*}B$ gives the
representatives

(4.1) |  |  |
---|---|---|---
$\textstyle{\pi^{*}E_{1}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\pi^{*}d^{*}}$$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{*}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}t^{\prime}}$for$\textstyle{\pi^{*}E_{\bullet}^{\vee}[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
\
\mathbb{T}_{\mathbf{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
\pi^{*}\mathbb{T}_{\mathbf{P}}}$$\textstyle{\pi^{*}E_{0}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,1)}$$\textstyle{\pi^{*}(B\oplus
E_{0}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,0)}$$\textstyle{\pi^{*}B}$

because the connecting homomorphism of the horizontal triangle of vertical
2-term complexes represents
$\pi^{*}\mathbb{T}_{\mathbf{P}}\to\mathbb{T}_{\mathbf{F}/\mathbf{P}}[1]=\pi^{*}E_{\bullet}^{\vee}$
by construction.

The zero section $\mathbf{P}\subset\mathbf{F}$ defines a splitting
$\mathbb{T}_{\mathbf{P}}\to\mathbb{T}_{\mathbf{F}}|_{P}$ of the triangle
$\mathbb{T}_{\mathbf{F}}\to\pi^{*}\mathbb{T}_{\mathbf{P}}\to\mathbb{T}_{\mathbf{F}/\mathbf{P}}[1]$
on $P$. We note for later that following through the above construction then
shows that on restriction to $P\subset F$, (4.1) splits as

(4.2)
$\textstyle{E_{1}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,0)}$$\scriptstyle{-d^{*}}$$\textstyle{E_{1}^{*}\oplus
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,1)}$$\scriptstyle{-d^{*}\oplus\\!}$$\scriptstyle{t^{\prime}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}}$$\textstyle{E_{0}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,0)}$$\textstyle{E_{0}^{*}\oplus
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,1)}$$\textstyle{B.\\!\\!}$

### Representative for $\mathbb{T}_{\mathbf{M}}$

Consider the diagram of exact triangles

(4.3)
$\textstyle{B|_{M}[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{1}}$$\textstyle{\mathbb{T}_{\mathbf{P}}|_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds}$$\textstyle{A|_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds_{0}}$$\textstyle{E_{1}|_{M}[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\bullet}|_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{0}|_{M}.\\!\\!}$

Since $\operatorname{Ext}^{1}(B|_{M},E_{0}|_{M})=0$ there exists a map
$s_{1}$; taking cones then gives the map marked $ds_{0}$. (Due to the choices
involved $ds_{0}$ is not entirely determined by $s_{0}$ — the composition of
$s$ with $E_{\bullet}\to E_{0}$ — so the notation is only suggestive.) Hence
we get a representative of $\mathbb{T}_{\mathbf{M}}$ — the cocone of
$\mathbb{T}_{\mathbf{P}}|_{M}\to E_{\bullet}^{\vee}|_{M}$ — as the total
complex of

$\textstyle{A|_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\prime}}$$\scriptstyle{ds_{0}}$$\textstyle{B|_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{1}}$$\textstyle{E_{0}|_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{E_{1}|_{M}.\\!}$

Since $\mathbf{M}$ is quasi-smooth $(s_{1},-d)\colon(B\oplus E_{0})|_{M}\to
E_{1}|_{M}$ is onto. Letting $K$ denote its kernel we get three quasi-
isomorphic representatives of $\mathbb{T}_{\mathbf{M}}$,

(4.4) $\displaystyle\mathbb{T}_{\mathbf{M}}$ $\displaystyle\cong$
$\displaystyle\big{\\{}A|_{M}\xrightarrow{\ (t^{\prime},ds_{0})\ }(B\oplus
E_{0})|_{M}\xrightarrow{\ (s_{1},-d)\ }E_{1}|_{M}\big{\\}}$ (4.5)
$\displaystyle\cong$ $\displaystyle\big{\\{}A|_{M}\xrightarrow{\
(t^{\prime},ds_{0})\ }K\big{\\}}$ (4.6) $\displaystyle\cong$
$\displaystyle\big{\\{}(A\oplus E_{1}^{*})|_{M}\xrightarrow{\
(t^{\prime},ds_{0})\oplus\operatorname{id}\ }K\oplus E_{1}^{*}|_{M}\big{\\}}.$

Since $\mathbb{T}^{\vee}_{\mathbf{M}}|_{M}\to\mathbb{L}_{M}$ is a perfect
obstruction theory, the Behrend-Fantechi construction [BF] applied to the
third complex (4.6) defines a cone

(4.7) $C_{M}\ \subset\ K\oplus E_{1}^{*}|_{M}\quad\text{such
that}\quad[\mathbf{M}]^{\operatorname{vir}}\ =\ 0^{\hskip 0.6pt!}_{K\oplus
E_{1}^{*}|_{M}}[C_{M}].$

For later we note that by (4.3) the map
$\mathbb{T}_{\mathbf{M}}|_{M}\to\mathbb{T}_{\mathbf{P}}|_{M}$ becomes the
projection of the complex (4.4) to its first two terms $t^{\prime}\colon
A|_{M}\to B|_{M}$. Thus it is also represented by the chain map from (4.5) to
$A|_{M}\to B|_{M}$ given by the identity on $A|_{M}$ and the composition
$K\hookrightarrow(B\oplus E_{0})|_{M}\to B|_{M}$ on $K$.

We further compose this with the (restriction to $M$ of the) map
$\mathbb{T}_{\mathbf{P}}\to\mathbb{T}_{\mathbf{F}}|_{P}$ induced by the zero
section $\mathbf{P}\subset\mathbf{F}$, to get a description of the map

(4.8)
$\mathbb{T}_{\mathbf{M}}|_{M}\longrightarrow\mathbb{T}_{\mathbf{F}}|_{M}$

induced by $\mathbf{M}\subset\mathbf{F}$. By (4.2)
$\mathbb{T}_{\mathbf{P}}\to\mathbb{T}_{\mathbf{F}}|_{P}$ is the inclusion of
$t^{\prime}\colon A\to B$ as the second summand of the central vertical
complex in (4.2). Thus (4.8) maps (4.5) in the obvious way to the second
summand of

(4.9)
$\textstyle{E_{1}^{*}|_{M}}$$\textstyle{E_{0}^{*}|_{M}}$$\textstyle{\mathbb{T}_{\mathbf{F}}\qquad\cong}$$\textstyle{\quad\oplus\quad\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-d^{*}}$$\scriptstyle{\oplus
t^{\prime}}$$\textstyle{\quad\oplus\quad}$$\textstyle{A|_{M}}$$\textstyle{B|_{M}.\\!\\!}$

### Representative for $\mathbb{T}_{T^{*}[-2]\mathbf{M}}$

On restriction to $M$ we have the splitting666The derivative of the zero
section $\mathbf{M}\to T^{*}[-2]\mathbf{M}$ splits the pullback to the zero
section of the exact triangle
$\mathbb{T}_{\mathbf{M}}^{\vee}[-2]\to\mathbb{T}_{T^{*}[-2]\mathbf{M}}\to\mathbb{T}_{\mathbf{M}}$.
$\mathbb{T}_{T^{*}[-2]\mathbf{M}}\cong\mathbb{T}_{\mathbf{M}}\oplus\mathbb{T}^{\vee}_{\mathbf{M}}[-2]$,
represented by the complex $\eqref{TM1}\hskip 0.6pt\oplus\hskip
0.6pt\eqref{TM1}^{\vee}[-2]$,

(4.10) $\quad(A\oplus
E_{1}^{*})\big{|}_{M}\xrightarrow[(-d^{*}\\!,\,s_{1}^{*})]{(t^{\prime}\\!,\,ds_{0})\oplus}(B\oplus
E_{0})\big{|}_{M}\oplus(E_{0}^{*}\oplus
B^{*})\big{|}_{M}\longrightarrow(A^{*}\oplus E_{1})\big{|}_{M}.$

Here the second arrow is the dual of the first. In fact (4.10) is also equal
to $\eqref{TM3}\hskip 0.6pt\oplus\hskip 0.6pt\eqref{TM3}^{\vee}[-2]$ because
$K\hookrightarrow(B\oplus E_{0})|_{M}$ induces $E_{1}^{*}\cong
K^{\perp}\hookrightarrow(E_{0}^{*}\oplus B^{*})|_{M}$ and hence an isomorphism
between $(K\oplus K^{\perp})\oplus(K\oplus K^{\perp})^{*}$ and $(B\oplus
E_{0}\oplus E_{0}^{*}\oplus B^{*})|_{M}$.

Thus the first arrow of (4.10) factors through the maximal isotropic subbundle
$K\oplus E_{1}^{*}|_{M}$. So when we consider the stupid truncation of (4.10)

(4.11) $\mathbb{T}_{\tau\mathbf{M}}\ :=\ \big{\\{}(A\oplus
E_{1}^{*})\big{|}_{M}\longrightarrow(B\oplus
E_{0})\big{|}_{M}\oplus(E_{0}^{*}\oplus B^{*})\big{|}_{M}\big{\\}}$

as defining a perfect obstruction theory
$\mathbb{T}_{\tau\mathbf{M}}^{\vee}\big{|}_{M}\to\mathbb{L}_{M}$ for $M$, the
induced Behrend-Fantechi cone is

(4.12) $C_{M}\,\stackrel{{\scriptstyle\eqref{CM}}}{{\subseteq}}\,K\oplus
E_{1}^{*}|_{M}\ \subset\ (B\oplus E_{0}\oplus E_{0}^{*}\oplus B^{*})|_{M}.$

The virtual cycle of $T^{*}[-2]\hskip 0.6pt\mathbf{M}$ is defined in [OT1,
Section 3.3]777This definition requires a choice of orientation on $(B\oplus
E_{0}\oplus E_{0}^{*}\oplus B^{*})|_{M}$. We use the choice $o_{K\oplus
E_{1}^{*}|_{M}}$ which makes $K\oplus E_{1}^{*}|_{M}$ a _positive_ maximal
isotropic. via this stupid truncation as

(4.13) $\sqrt{0^{\,!}}_{(B\oplus E_{0}\oplus E_{0}^{*}\oplus
B^{*})|_{M}}[C_{M}]\ =\ 0^{\,!}_{K\oplus E_{1}^{*}|_{M}}[C_{M}]\
\stackrel{{\scriptstyle\eqref{CM}}}{{=}}\ [\mathbf{M}]^{\operatorname{vir}},$

where the first equality is [OT1, Lemma 3.5] applied to the isotropic
embedding (4.12).

### Relating $\mathbf{M}$ and $\mathbf{F}$

The stupid (dual) perfect obstruction theory (4.11) sits inside the exact
triangle

$\textstyle{(E_{0}\oplus
B^{*})\big{|}_{M}[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hskip
51.21495pt(E_{0}\oplus
B^{*})\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{\tau\mathbf{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{=}$$\textstyle{(A\oplus
E_{1}^{*})\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(B\oplus
E^{*}_{0}\oplus E_{0}\oplus
B^{*})\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{\mathbf{F}}\big{|}_{M}}$$\textstyle{(A\oplus
E_{1}^{*})\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(B\oplus
E_{0}^{*})\big{|}_{M}.\hskip 51.21495pt}$ (4.14)

On the bottom row we have used the splitting (4.2), which also shows the
obvious vertical arrows are chain maps.

We next show this fits into the commutative diagram of perfect obstruction
theories (4.15) below. By (4.8, 4.9)
$\mathbb{T}_{\mathbf{M}}\to\mathbb{T}_{\mathbf{F}}|_{M}$ factors through the
above map $\mathbb{T}_{\tau\mathbf{M}}\to\mathbb{T}_{\mathbf{F}}|_{M}$ by the
natural inclusion of (4.5) into (4.11). This defines the top row of the
commutative diagram

$\textstyle{\mathbb{T}^{\vee}_{\mathbf{F}}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{T}_{\tau\mathbf{M}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}^{\vee}_{\mathbf{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{F}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{M},\\!\\!}$

with the vertical maps induced by $F\hookrightarrow\mathbf{F}$ and
$M\hookrightarrow\mathbf{M}$ respectively. Combining the left hand square with
the dual of (4.14) gives the exact triangle of perfect obstruction theories

(4.15) $\textstyle{(E_{0}^{*}\oplus
B)\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(ds_{0})^{*}}$$\scriptstyle{\oplus
s_{1}}$$\textstyle{\mathbb{T}^{\vee}_{\mathbf{F}}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{T}_{\tau\mathbf{M}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(E_{0}^{*}\oplus
B)\big{|}_{M}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{M/F}[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{F}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{M/F}.\\!\\!}$

### Isotropic rational equivalence of cones

Applying the Behrend-Fantechi construction to the perfect obstruction theory
(4.1) for $F$ gives a cone

(4.16) $C_{F}\ \subset\ \pi^{*}(B\oplus E_{0}^{*})\ =:\ \Lambda^{*}\ \subset\
\Lambda\oplus\Lambda^{*}.$

Then by [KKP] the exact triangle (4.15) — and its realisation (4.14) as a
short exact sequence of 2-term chain complexes of locally free sheaves —
induces a rational equivalence between

$C_{M}\ \subset\ (\Lambda\oplus\Lambda^{*})\big{|}_{M}\ \text{ and }\
C_{M/C_{F}}\ \subset\ (\Lambda\oplus\Lambda^{*})\big{|}_{M}\,.$

In order to prove this rational equivalence takes places inside an isotropic
substack (in fact cone) of $(\Lambda\oplus\Lambda^{*})\big{|}_{M}$ we review
parts of the [KKP] construction. For more details see also [BCM, Ma].

Fix any Deligne-Mumford stacks $M\subset F$ with perfect obstruction theories
$\mathbb{T}_{M}^{\vee}\to\mathbb{L}_{M},\
\mathbb{T}_{F}^{\vee}\to\mathbb{L}_{F}$ fitting into a diagram of exact
triangles

$\textstyle{\Lambda[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{F}|_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda}$$\textstyle{\mathbb{L}_{M/F}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{M}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{F}^{\vee}|_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{M/F}^{\vee}[1],\\!\\!\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

with the (first three terms of the) top row represented by a short exact
sequence of (vertical) 2-term complexes of vector bundles

(4.17)
$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\Lambda^{*}.\\!\\!}$

For us these will be provided by (4.14) and (4.15) with $\mathcal{A}=(A\oplus
E_{1}^{*})|_{M},\ \Lambda=(B^{*}\oplus E_{0})|_{M}$ and
$E=\Lambda\oplus\Lambda^{*}$.

Using this data [KKP, Proposition 1] defines a canonical abelian cone stack (a
certain normal sheaf) inside a bundle stack

$[N/\mathcal{A}]\ \subset\
[E_{t}/\mathcal{A}]\,\longrightarrow\,M\times\mathbb{C}.$

Here $E_{t}$ is the bundle over $M\times\mathbb{C}$ given by degenerating the
extension $E$ to its splitting $\Lambda\oplus\Lambda^{*}$; over
$t\in\mathbb{C}$ it is the kernel of
$(t\operatorname{id},\,\pi)\colon\Lambda^{*}\oplus E\to\to\Lambda^{*}$. (In
our situation $E=\Lambda\oplus\Lambda^{*}$ is already split so $E_{t}$ is just
the pullback of $E$ from $M$ to $M\times\mathbb{C}$.) We have also pulled
$\mathcal{A}$ back to $M\times\mathbb{C}$.

Then [KKP, Equation (8)] defines a canonical normal cone stack
$[C/\mathcal{A}]\subset[N/\mathcal{A}]$ containing a rational equivalence
between any $t\neq 0$ fibre $[C_{M}/\mathcal{A}]$ — the intrinsic normal cone
of $M$ — and the central fibre $[C_{M/C_{F}}/\mathcal{A}]$ over $t=0$. Pulling
back by $E_{t}\to[E_{t}/\mathcal{A}]$ gives, in our situation, a canonical
cone $C\subset\mathrm{tot}_{M\times\mathbb{C}}(\Lambda\oplus\Lambda^{*})$
satisfying

  * •

the fibre of $C$ over points $i_{t}\colon\\{t\\}\hookrightarrow\mathbb{C}$
with $t\neq 0$ is $C_{M}$,

  * •

the fibre of $C$ over $t=0$ _contains_ $C_{M/C_{F}}$, and

  * •

$i_{0}^{\hskip 0.6pt!}[C]=[C_{M/C_{F}}]$.

This gives our rational equivalence between $i_{1}^{\hskip 0.6pt!}[C]=[C_{M}]$
and $[C_{M/C_{F}}]$ inside $C\subset(\Lambda\oplus\Lambda^{*})|_{M}$.

About any point of $M$ our (exact triangle of) perfect obstruction theories
(4.14, 4.15) is isomorphic to one arising from the local model (2.5).888After
shrinking $M$ and $P$ we can find a local Kuranishi structure $(A,B,t)$ for
$P$ which induces the perfect obstruction theory $dt|_{P}=t^{\prime}\colon
A|_{P}\to B|_{P}$; see [OT2, Theorem 3.3] for example. (Here $A$ may be taken
to be an open set in a vector space, excusing our abuse of notation in
identifying it with its tangent spaces $A$. We are also using $B$ to denote
both the bundle on $P$ and a choice of a local extension of it to $A$.) Then
pick local lifts of $d\colon E_{0}\to E_{1}$ and $s$ to $A$ and proceed as in
(2.5) and Section 3 in the ambient space
$\mathcal{A}=\operatorname{tot}_{A}(E_{1}^{*})\xrightarrow{\ p\ }A$. This
gives Kuranishi charts (local over $M$ but global in the $p$-fibre directions)
$(\mathcal{A},\Lambda\oplus\Lambda^{*},\sigma)$ for $\tau\hskip 0.6pt\bf M$
(3.5) and $(\mathcal{A},\Lambda^{*},\sigma_{2})$ for $\bf F$ (3.6), compatible
under the projection $\Lambda\oplus\Lambda^{*}\to\Lambda^{*}$ which maps
$\sigma=(\sigma_{1},\sigma_{2})$ (3.4) to $\sigma_{2}$ (3.6). Taking their
derivatives along $M$ to pass back to perfect obstruction theories recovers
precisely (4.14, 4.15) and so the exact triangle (4.17) required to apply
[KKP]. In this local model [KKP]’s cone $C$ is, by construction, the cone
$\overline{\Gamma}\times_{\mathbb{C}^{2}}(w=0)$ (3.10) of Remark 3.9.

In this local model $C_{M}$ — the limit of the isotropic graphs (3.5) — and
$C$ — the family over $(w=0)$ in Remark 3.9 — are _isotropic_ in
$\mathrm{tot}_{M\times\mathbb{C}}(\Lambda\oplus\Lambda^{*})$. Since $C$ is
canonical it is isomorphic to its local model and is therefore also isotropic.

Therefore we can apply the deformation invariance [OT1, Equation (78)] of
$\surd\hskip 0.6pt0^{\hskip 0.6pt!}$ to (4.13) to give

(4.18) $[\mathbf{M}]^{\operatorname{vir}}\ =\
(-1)^{|\Lambda|+\operatorname{rank}\Lambda}\sqrt{0^{\,!}_{\Lambda\oplus\Lambda^{*}}}\,[C_{M/C_{F}}].$

Finally we want to replace $C_{M/C_{F}}$ by $C_{F}$ by deforming the embedding
$(0,c)\colon C_{F}\hookrightarrow\Lambda\oplus\Lambda^{*}$ of (4.16), where
$c$ is the embedding $C_{F}\hookrightarrow\Lambda^{*}$.

Writing $\mathbb{T}_{\mathbf{F}}=\\{\mathcal{A}\to\Lambda^{*}\\}$ as in (4.1)
we can consider the composition

$\Lambda^{*}\longrightarrow h^{1}(\mathbb{T}_{\mathbf{F}})\xrightarrow{\
h^{1}(d\hskip 0.6pt\tilde{s})\ }\mathcal{O}_{F}$

as a section of $\Lambda$ which we also denote by $h^{1}(d\hskip
0.6pt\tilde{s})$. We use this to perturb $(0,c)$ (4.16), taking the closure of
the graph

(4.19) $\overline{\Gamma_{(w^{-1}h^{1}(d\hskip 0.6pt\tilde{s}),\,c)}}\
\subset\ \mathrm{tot}_{F\times\mathbb{P}^{1}}(\Lambda\oplus\Lambda^{*}),$

where $w$ is the coordinate pulled back from $\mathbb{P}^{1}$. Because
$h^{1}(d\hskip 0.6pt\tilde{s})$ cuts out $M\subset C_{F}$ this gives the
standard (flat!) deformation of $C_{F}$ to the normal cone of $M\subset
C_{F}$. It therefore gives a rational equivalence between
$C_{M/C_{F}}\subset(\Lambda\oplus\Lambda^{*})|_{M}$ and the fibre (4.16) over
$w=\infty$.

Since $h^{1}(d\hskip 0.6pt\tilde{s}))$ is a cosection the composition

$C_{F}\,\lhook\joinrel\relbar\joinrel\rightarrow\,\Lambda^{*}\xrightarrow{\
h^{1}(d\hskip 0.6pt\tilde{s})\ }\mathcal{O}_{F}$

vanishes, which means the rational equivalence (4.19) is _isotropic_. So by
the deformation invariance [OT1, Equation (78)] again, (4.18) has become

$[\mathbf{M}]^{\operatorname{vir}}\ =\
(-1)^{|\Lambda|+\operatorname{rank}\Lambda}\sqrt{0^{\,!}_{\Lambda\oplus\Lambda^{*}}}\hskip
0.6pt[C_{F}],$

where $C_{F}$ is embedded in $\Lambda\oplus\Lambda^{*}$ via $(h^{1}(d\hskip
0.6pt\tilde{s}),c)$. Finally, this class is defined in [OT1, Section 3.2] to
be the intersection of $C_{F}$ with the 0-section of $\Lambda^{*}$, cosection
localised by the tautological cosection $\tau_{\Lambda}\big{|}_{F}$ of the
pullback of $\Lambda$ to $F$. But since $F$ is embedded in this pullback by
the graph of $h^{1}(d\hskip 0.6pt\tilde{s})$, this cosection is just
$h^{1}(d\hskip 0.6pt\tilde{s})$, yielding

$[\mathbf{M}]^{\operatorname{vir}}\ =\
(-1)^{|\Lambda|+\operatorname{rank}\Lambda}\,0^{\,!,\,\mathrm{loc}}_{\Lambda^{*},\,h^{1}(d\hskip
0.6pt\tilde{s})}[C_{F}]\ =\ (-1)^{e}[\mathbf{F}]^{\mathrm{loc}}.$

The verification of the sign
$(-1)^{|\Lambda|+\operatorname{rank}\Lambda}=(-1)^{e}$ was done in (3.8).

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<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Department of Mathematics
Imperial College London
London SW7 2AZ
United Kingdom


# Ladder-of-Thought: Using Knowledge as Steps to Elevate Stance Detection

###### Abstract

Stance detection aims to identify the attitude expressed in a document towards
a given target. Techniques such as Chain-of-Thought (CoT) prompting have
advanced this task, enhancing a model’s reasoning capabilities through the
derivation of intermediate rationales. However, CoT relies primarily on a
model’s pre-trained internal knowledge during reasoning, thereby neglecting
the valuable external information that is previously unknown to the model.
This omission, especially within the unsupervised reasoning process, can
affect the model’s overall performance. Moreover, while CoT enhances Large
Language Models (LLMs), smaller LMs, though efficient operationally, face
challenges in delivering nuanced reasoning. In response to these identified
gaps, we introduce the Ladder-of-Thought (LoT) for the stance detection task.
Constructed through a dual-phase Progressive Optimization Framework, LoT
directs the small LMs to assimilate high-quality external knowledge, refining
the intermediate rationales produced. These bolstered rationales subsequently
serve as the foundation for more precise predictions - akin to how a ladder
facilitates reaching elevated goals. LoT achieves a balance between efficiency
and performance. Our empirical evaluations underscore LoT’s efficacy, marking
a $16\%$ improvement over GPT-3.5 and a $10\%$ enhancement compared to GPT-3.5
with CoT on stance detection task.

Index Terms—  Stance Detection, Ladder-of-Thought, Language Model, Knowledge
Infusion

##  1 Introduction

Stance detection is the task of discerning the stance towards a specific
target in an provided document. This task can be challenging given the breadth
of topics and the depth of reasoning required to make accurate predictions.
Nevertheless, the landscape of stance detection has evolved significantly with
the success of Pre-trained Language Models (PLMs). These PLMs, when fine-tuned
for downstream tasks, demonstrate a remarkable improvement in performance [1,
2].

Paradigms | Knowledge | Sizes | Reasoning | Performance
---|---|---|---|---
WS-BERT | External | 340M | Weak | 74.5
CoT | Internal | 175B | Strong | 68.9
LoT (ours) | External | 780M | Strong | 79.2
Table 1: Comparison of different stance detection paradigms.

Leveraging the capabilities of LMs, prompt-based techniques have further
enhanced the performance, especially when LLMs such as GPT-3.5 are equipped
with meticulously designed prompts [3]. The Chain-of-Thought (CoT) prompting
stands as a prominent prompting strategy, enabling LMs to produce coherent and
systematic reasoning rationales, which in turn improves the subsequent
prediction accuracy [4]. However, CoT has a discernible limitation: it mainly
relies on the model’s internal, pre-existing knowledge when generating these
rationales [5]. External knowledge, which is often dynamic, evolving, and
abundant in domain-specific insights, remains unexploited [6]. Given CoT’s
reliance on the model’s pre-trained knowledge, its unsupervised intermediate
reasoning process may inevitably produce less reliable rationales, affecting
the model’s overall performance [5, 6, 7, 8].

The integration of external background knowledge is paramount for optimizing
models’ stance detection capabilities [9]. Predictions can be compromised in
the absence of this auxiliary information, particularly when limited by the
model’s intrinsic knowledge. Table 1 serves as a testament: despite ChatGPT’s
utilization of CoT [3], smaller models like BERT can outperform it in stance
detection tasks when supplemented with external knowledge from Wikipedia [9].

Moreover, the expansive architecture of LLMs like GPT-3.5 brings concerns
about efficiency. On the other hand, smaller LMs, though more operationally
efficient, often compromise on the reasoning capability due to their
compactness [4, 7]. And while CoT provides performance gain in LLMs, it does
not effectively benefit the smaller-sized models [4]. This underscores the
need for enhancing the reasoning prowess of smaller models without bloating
their size.

To address these challenges, we propose Ladder-of-Thought (LoT), a novel
methodology that leverages external knowledge as steps to elevate stance
detection. LoT operates on a Progressive Optimization Framework. The ”ladder”
in LoT represents this Progressive Optimization Process. The initial phase
absorbs external information, guiding the model to generate more reliable
intermediate knowledge as rationales. This intermediate knowledge, act as
”steps” to progressively elevate the model’s comprehensive understanding
capability, culminating in a robust stance detection. Tailored for smaller
LMs, LoT strikes a harmonious balance between efficiency and performance. It
facilitates the seamless integration of ample external knowledge and
cultivates profound reasoning capabilities. The architecture of LoT is
illustrated in Fig. 1.

Our main contributions are summarized as follows:

  * •

We introduce LoT – a novel method for stance detection. By enriching smaller
LMs with external knowledge, LoT effectively facilitates the generation of
more reliable intermediate rationales, consequently enhancing the prediction
performance.

  * •

We demonstrate that LoT outperforms existing methods, achieving state-of-the-
art results while maintaining efficiency.

Fig. 1: The Overview of Ladder-of-Thought Architecture

##  2 Methodology

###  2.1 Task Definition:

Stance Detection: Stance detection involves identifying the stance of an
opinionated document concerning a specific target. Formally, consider a set
$D=\\{(x_{i}=(d_{i},t_{i}),y_{i})\\}_{i=1}^{n}$ representing $n$ instances.
Here, $x_{i}$ encapsulates a document $d_{i}$ and a target $t_{i}$. The task
is to deduce the stance label, $y_{i}$, which can be categorized as
$\\{\text{positive, negative, neutral}\\}$.

###  2.2 External Knowledge Retrieval

To increase the reliability of the generated intermediate rationales in LoT,
we integrate external knowledge to enhance the generation in a supervised
manner. Specifically, a web retrieval process fetches pertinent external
information for each target $t_{i}$ from Google Search. By extending beyond
the traditional realms of Wikipedia and diving into the wider web, we access a
plethora of diverse and dynamic information [10]. This shift aligns with the
emerging trend of exploring beyond the boundaries of Wikipedia-based research
[11, 12, 10].

###  2.3 Ladder-of-Thought (LoT) Architecture:

The Ladder-of-Thought (LoT) architecture enhances stance detection, enabling
smaller models to reason more effectively. LoT draws its metaphor from the
construction of a ladder, where the process of Progressive Optimization forms
the framework of the ladder, and the reliable intermediate knowledge,
fortified with external insights, serves as the integral ”steps”. These
pivotal steps empower the model to reach heightened insights and deeper
comprehension, facilitating more accurate predictions. LoT is developed
through a dual-phase Progressive Optimization Framework:

  1. 1.

Phase 1 - Generation Fine-tuning: In this foundational phase, the pre-trained
model $M_{0}$ is fine-tuned with the retrieved knowledge. This transfers the
external insights to the model, guiding it to generate more robust
intermediate knowledge that subsequently aids in downstream stance
predictions. The resulting model $M_{1}$ facilitates the generation of more
enriched and reliable intermediate rationales, denoted as
$k_{\textit{intermediate\\_i}}$.

  2. 2.

Phase 2 - Prediction Fine-tuning: Phase-2 utilizes the enhanced knowledge
generated from Phase-1 to expertly discern stance labels. By concatenating the
document, target, and the generated knowledge, we construct an enhanced input
representation, $x_{\textit{enhanced\\_i}}$. $M_{1}$ is then fine-tuned with
this enhanced input, culminating in the final model $M_{2}$. Given the
knowledge-infused input, $M_{2}$ can conduct stance prediction $y_{i}$.

The Ladder-of-Thought (LoT) architecture employs a Progressive Optimization
Framework to enhance the stance detection model step-by-step. Leveraging the
concept of cognitive evolution, LoT signifies a novel paradigm for model
training. In particular, phase-1 is the foundation of LoT, infusing the model
with core knowledge, reminiscent of grounding a student in fundamental
theories. In Phase-2, this grounded rationale is utilized to guide the model
towards more nuanced stance detection. The optimization from $M_{0}$ to
$M_{2}$ via $M_{1}$ reflects the LoT philosophy: evolving model capabilities
through deliberate optimization, striking a balance between computational
efficiency and reasoning depth.

For a detailed step-by-step procedure of the Progressive Optimization, refer
to Algorithm 1.

Algorithm 1 Progressive Optimization Algorithm

1:Document matrix $D=\\{d_{1},d_{2},...,d_{n}\\}$, Target vector
$T=\\{t_{1},t_{2},...,t_{n}\\}$, Pre-trained model $M_{0}$

2:Stance prediction vector $Y=\\{y_{1},y_{2},...,y_{n}\\}$

3:function LoT($D,T,M_{0}$)

4: Phase-1:

5: for $i=1$ to $n$ do

6: $k_{i}\leftarrow\text{WebRetrieval}(t_{i})$

7:
$M_{1}\leftarrow\text{GenerationFinetune}(\\{k_{1},k_{2},...,k_{n}\\},M_{0})$

8: for $i=1$ to $n$ do

9: $k_{\textit{intermediate\\_i}}\leftarrow M_{1}(d_{i},t_{i})$

10:
$x_{\textit{enhanced\\_i}}\leftarrow\text{IntegrateInputs}(d_{i},k_{\textit{intermediate\\_i}},t_{i})$

11: Phase-2:

12:
$M_{2}\leftarrow\text{PredictionFinetune}(\\{x_{\textit{enhanced\\_1}},x_{\textit{enhanced\\_2}},...,$

13: $x_{\textit{enhanced\\_n}}\\},M_{1})$

14: for $i=1$ to $n$ do

15: $y_{i}\leftarrow M_{2}(x_{\textit{enhanced\\_i}})$

16: return $Y$

##  3 Experiment

###  3.1 Dataset and Evaluation Metric

The VAried Stance Topics (VAST) [13] is a classic zero-shot and few-shot
stance detection dataset. It encompasses a broad spectrum of topics: 4,003 for
training, 383 for development, and 600 for testing. Unlike other datasets for
stance detection like P-stance [14] which only have 2 targets or SemEval-2016
[15] with 4 targets, VAST covers a numerous number of targets spanning various
domains. Following the preceding studies [13, 9], the macro average of
F1-score is used as the evaluation metric.

###  3.2 Baselines and Models

We employ FLAN-T5-Large, the 780M parameter version of FLAN-T5, as our
backbone. We compare our model with the following baselines: TGA-Net [13],
BERT, BERT-GCN [16], CKE-Net [2], WS-BERT-Single [9], DQA [3], StSQA [3]. The
first five methods are based on BERT and its variants. DQA is based on ChatGPT
with direct input-output (IO) prompting, while StSQA employs CoT on ChatGPT,
prompting ChatGPT in a step-by-step manner.

###  3.3 Result

The overall results of our model and the baselines are reported in Table 2.

Methods | Models | F1 Scores
---|---|---
TGA-Net | BERT | 66.5
BERT | BERT | 68.4
BERT-GCN | BERT | 69.2
CKE-Net | BERT | 70.1
WS-BERT-Single | BERT | 74.5
DQA | GPT-3.5 | 62.3
StSQA | GPT-3.5 | 68.9
Baseline FLAN-T5 | FLAN-T5 | 73.6
LoT (Ours) | FLAN-T5 | 79.2
Table 2: Performance comparison on the VAST dataset.

Compared to the baseline FLAN-T5, LoT achieves a remarkable improvement,
achieving an F1 score of $79.2$, while FLAN-T5 achieves an F1 score of $73.6$.
This highlights the efficacy of our LoT. Furthermore, compared to ChatGPT-
based DQA, which operates on an expansive architecture and achieves an F1
score of $62.3$, our LoT demonstrates not just superior performance but
tangible efficiency with significantly fewer parameters. This compact model
size promises better deployment possibilities in real-world scenarios where
computational resources can be a constraint.

Compared to StSQA with an F1 score of $68.9$, our LoT also outperforms this
CoT-enhanced ChatGPT approach. This result showcases that despite CoT
amplifying internal reasoning, our LoT can absorb high-quality external
knowledge, facilitating a more accurate prediction.

###  3.4 Ablation Study

The foundational structure of LoT is built on the dual-phase Progressive
Optimization framework. As all implementations involves the Prediction Fine-
tuning, our focus lies in understanding the efficacy of two specific aspects
of LoT: Generation Fine-tuning and the enhanced intermediate knowledge. We
conduct an ablation study to evaluate their individual and comprehensive
impact. In addition to the baseline and the complete LoT implementation, we
introduce two intermediate settings for comprehensive comparison:

CoT: Following the principle of CoT, this configuration skips Generation Fine-
tuning, directly utilizing the pre-trained model to produce intermediate
knowledge and perform the subsequent prediction. This offers insights into the
impact of the raw knowledge that is directly prompted from the pre-trained
model on prediction performance.

Phase1-Only: Focusing exclusively on Phase-1 Fine-tuning, this configuration
omits the subsequent integration of the generated knowledge during Phase-2
fine-tuning. The objective is to evaluate the direct influence of Phase-1
Fine-tuning and determine if it enhances the model’s intrinsic knowledge.

Models | Gen Fine-tuning | Gen Knowledge | F1
---|---|---|---
Baseline | – | – | 73.4
CoT | – | ✓ | 73.1
Phase1-Only | ✓ | – | 74.2
LoT | ✓ | ✓ | 79.2
Table 3: Ablation study on LoT.

Table 3 showcases the results of the ablation study.

The Baseline achieves an F1 score of 73.4, representing the performance
without any additional enhancements.

By comparison, the CoT configuration slightly decreases to 73.1. This aligns
with our prior discussion that small models may not benefit from CoT due to
their limited reasoning capabilities [4]. Although directly prompting for
intermediate knowledge yields some rationales, their quality is compromised.
The unsupervised nature of these intermediate outputs may introduce potential
noise. Hence, introducing CoT might inadvertently add complexity to the
models, distracting them from accurate prediction. This underscores the
significance of a supervised fine-tuning phase to enhance the reliability in
knowledge generation.

The Phase1-Only configuration achieves an F1 score of 74.2, surpassing our
baseline. This score suggests that Generation Fine-tuning can effectively
enhance the model’s inherent knowledge base. By supplementing the model with
external information, even without explicitly leveraging the generated
knowledge during predictions, we can still witness an improvement over the
baseline performance. This underscores that enriching the foundational
knowledge of the model can inherently bolster its capabilities in stance
detection.

With our LoT configuration, the model reaches an F1 score of 79.2, showcasing
a remarkable performance improvement over both the baseline and the other
configurations. This substantial increase underlines the benefits of our
overall Progressive Optimization Framework in LoT.

###  3.5 Overfitting in Progressive Optimization

In our Progressive Optimization Framework, overfitting presents a notable
challenge. If the model undergoes excessive training during Generation Fine-
tuning (Phase-1), it could become overly specialized for the initial task,
leading to a detrimental impact on its performance during the subsequent
predictions. Achieving the ideal equilibrium between these phases is crucial.
We investigate the influence of training epochs in Phase-1 on the subsequent
prediction accuracy in Phase-2. The outcomes are depicted in Figure 2.

Fig. 2: Effect of Phase-1 Training epochs on the overall prediction accuracy.

The findings suggest that the optimal performance is achieved at around 2
epochs, with a subsequent decline in performance as the number of epoch
increases. This juncture signifies the ideal balance: it facilitates the
generation of high-quality intermediate knowledge without an excessive
reliance on Phase-1. While Phase-1 aims to enhance the model’s reasoning for
Phase-2, it is important to avoid overemphasizing the former phase at the
expense of the latter. Our results highlight the importance of a strategic
equilibrium, ensuring that each phase complements the other, ultimately
constructing a robust and effective Progressive Optimization Framework.

##  4 Conclusion

In this research, we introduce the Ladder-of-thought (LoT). This method
effectively enhances the smaller LMs’ reasoning abilities with a dual-phase
Progressive Optimization Framework. LoT enables the model to efficiently
absorb high-quality external knowledge, thereby crafting more reliable
intermediate rationales that facilitate accurate predictions. Our empirical
evaluations demonstrate the efficacy of LoT, highlighting its superiority over
the existing methods. LoT showcases that even smaller LMs, with the right
guidance, can outperform LLMs like ChatGPT in stance detection. LoT is also
applicable to other downstream tasks, and we aim to explore further in future
works.

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