$\big{(}2M,C^{\infty}_{2M}\big{)}\hookrightarrow\big{(}\widetilde{\mathsf{K}}^{*},C^{\infty}_{\widetilde{\mathsf{K}}^{*}}\big{)}$
such that the composition $\mathcal{A}_{2}\subset
C^{\infty}_{\widetilde{\mathsf{K}}^{*}}\to\to C^{\infty}_{2M}$ has image
$\mathcal{O}_{2M}$ by (5.16) and Lemma 5.17. It has kernel $I_{1}^{2}$ by
construction (5.16).

Now we show that $I_{1}$ (5.12) arises in the way stipulated by Definition
1.16. As usual let $\mathcal{I}\subset\mathcal{O}_{2M}$ be the ideal of $M$,
so that

$I_{2}\ :=\ \ker\big{(}\mathcal{A}_{2}\longrightarrow C^{\infty}_{M}\big{)}\
=\
C^{\infty}_{\widetilde{\mathsf{K}}^{*}}\times_{C^{\infty}_{\widetilde{\mathsf{K}}^{*}}/I_{1}^{2}}\mathcal{I}$

and

$C^{\infty}_{\widetilde{\mathsf{K}}^{*}}.I_{2}\ =\
C^{\infty}_{\widetilde{\mathsf{K}}^{*}}\times_{C^{\infty}_{\widetilde{\mathsf{K}}^{*}}/I_{1}^{2}}\Big{(}\big{(}C^{\infty}_{\widetilde{\mathsf{K}}^{*}}/I_{1}^{2}\big{)}.\hskip
0.6pt\mathcal{I}\Big{)}.$

Locally, on $M_{i}$, we have $\mathcal{I}=J/J^{2}\oplus
K\subset\mathcal{O}_{(2M)_{\mathbb{P}^{N}}}\oplus K$ so we see from (5.12)
that $\big{(}C^{\infty}_{\widetilde{\mathsf{K}}^{*}}/I_{1}^{2}\big{)}.\hskip
0.6pt\mathcal{I}=I_{1}/I_{1}^{2}$. Since $\phi_{i}$ (5.14) clearly preserves
$I_{1}/I_{1}^{2}\subset C^{\infty}_{\widetilde{\mathsf{K}}^{*}}/I_{1}^{2}$ we
deduce that

$C^{\infty}_{\widetilde{\mathsf{K}}^{*}}.I_{2}\ =\
C^{\infty}_{\widetilde{\mathsf{K}}^{*}}\times_{C^{\infty}_{\widetilde{\mathsf{K}}^{*}}/I_{1}^{2}}\big{(}I_{1}/I_{1}^{2}\big{)}\
=\ I_{1}.$

Finally it remains to show that
$H\big{(}g_{1},\ldots,g_{n}\big{)}\in\mathcal{A}_{2}$ when
$g_{1},\ldots,g_{n}\in\mathcal{A}_{2}$ and
$H\colon\mathbb{C}^{n}\to\mathbb{C}$ is holomorphic on
$\operatorname{im}\big{(}$ $\times$ ${\\!}_{j}\hskip 0.6ptg_{j}\big{)}$. By
the definition of $\mathcal{A}_{2}$ (5.16), it is enough to show that the
projection of $H\big{(}g_{1},\ldots,g_{n}\big{)}$ to
$C^{\infty}_{\widetilde{\mathsf{K}}^{*}}/I_{1}^{2}$ lies in
$\phi\big{(}\mathcal{O}_{2M}\big{)}$.

Given $h_{1},\ldots,h_{n}\in I_{1}^{2}$, Hadamard’s Lemma gives

$H\big{(}g_{1}+h_{1},\ldots,g_{n}+h_{n}\big{)}-H\big{(}g_{1},\ldots,g_{n}\big{)}\
=\ \sum\nolimits_{i}a_{i}h_{i}\ \in\ I_{1}^{2},$

with no $\bar{h}_{i}$ terms because the antiholomorphic derivatives
$\overline{\partial}_{i}H$ vanish. Thus $H$ descends to a map

$\overline{H}\
\colon\Big{(}C^{\infty}_{\widetilde{\mathsf{K}}^{*}}/I_{1}^{2}\Big{)}^{\oplus
n}\longrightarrow C^{\infty}_{\widetilde{\mathsf{K}}^{*}}/I_{1}^{2},$

and we are left with showing that $\overline{H}$ preserves
$\phi(\mathcal{O}_{2M})\subset
C^{\infty}_{\widetilde{\mathsf{K}}^{*}}/I_{1}^{2}$.

This is a local property, and $\overline{H}$ obviously preserves
$\mathcal{O}_{2M}$, so it is sufficient to prove that it commutes with
$\phi_{i}$ over $2M_{i}$:

$\overline{H}\big{(}\phi_{i}(u_{1}),\ldots,\phi_{i}(u_{n})\big{)}\ =\
\phi_{i}\big{(}\hskip 0.6pt\overline{H}(u_{1},\ldots,u_{n})\big{)}.$

This follows from another application of Hadamard’s Lemma which, together with
the formula (5.14) for $\phi_{i}$, shows that, working modulo $I_{1}^{2}$,

$\displaystyle\overline{H}\big{(}\phi_{i}(u_{1}),\ldots,\phi_{i}(u_{n})\big{)}\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!H\big{(}u_{1},\ldots,u_{n}\big{)}+\sum\nolimits_{j}(\partial_{j}H)\big{(}u_{1},\ldots,u_{n}\big{)}.\big{\langle}\partial
u_{j},e_{i}\big{\rangle}$ $\displaystyle=$
$\displaystyle\\!\\!H\big{(}u_{1},\ldots,u_{n}\big{)}+\big{\langle}\partial\big{(}H(u_{1},\ldots,u_{n})\big{)},e_{i}\big{\rangle}$
$\displaystyle=$ $\displaystyle\\!\\!\phi_{i}\big{(}\hskip
0.6pt\overline{H}(u_{1},\ldots,u_{n})\big{)}.$

###  5.3. The global complex Kuranishi chart

We can now enhance $M$’s $\mathcal{A}_{2}$-embedding of Theorem 5.11 to a
global complex Kuranishi chart for $M$. Pick trivial algebraic bundles

(5.23) $E_{i}\,\text{ over the total space of }\,\widetilde{K}_{i}^{*}$

extending the trivial algebraic bundles $E_{1}|_{M_{i}}$, and algebraic
sections $s_{i}$ extending the section of $E_{1}\otimes\mathcal{I}|_{M_{i}}$
given by the central vertical arrow of (5.5). Restricting to open
neighbourhoods $V_{i}\subset\widetilde{K}_{i}^{*}$ of $M_{i}$ gives local
holomorphic Kuranishi charts

(5.24) $(V_{i},E_{i},s_{i}).$

We would like to $\mathcal{A}$-glue these to give a global chart. By Theorem
1.24 there exists an $\mathcal{A}_{1}$-bundle $E$ over some open neighbourhood
$V$ of $M\subset\widetilde{K}^{*}$ with $E|_{M}\cong E_{1}$ and which is
isomorphic over $V_{i}\cap V$ to
$E_{i}\otimes_{\mathcal{O}_{V_{i}}}\mathcal{A}_{1}$ (since it is trivial over
$V_{i}\cap V$). By shrinking $V$ and $V_{i}$ we may assume that
$V_{i}=\widetilde{K}_{i}^{*}\cap V$ and that the surjection
$E^{-1}\to\to\mathcal{I}$ on $M$ of (5.5) extends to a surjection of
$\mathcal{A}_{1}$-modules

$E^{*}\stackrel{{\scriptstyle s}}{{\longrightarrow\to\ }}I_{2}$

by Theorem 1.28. This gives our global complex Kuranishi chart $(V,E,s)$. By
construction and (5.10) the associated perfect obstruction theory

(5.25) $\textstyle{E^{*}\big{|}_{M}\cong
E^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds}$$\scriptstyle{s}$$\textstyle{\Omega_{V}\big{|}_{M}\cong
E^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}\big{/}I_{1}^{2}\cong\mathcal{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{V}\big{|}_{M}}$

is (5.5). This proves Theorem 2.

###  5.4. Purely holomorphic description

In sum, starting with the local holomorphic charts $(V_{i},E_{i},s_{i})$
(5.24), we have found $\mathcal{A}_{2}$-gluings $\psi_{ij}$ of the $V_{i}$ and
$\mathcal{A}_{1}$-gluings $\Phi_{ij}$ of the $E_{i}$. Furthermore we have a
section $s$ of the glued bundle $E$ whose restriction $s\colon
E^{*}|_{M}\to\mathcal{I}$ to $M_{i}$ is $s_{i}\colon
E_{i}^{*}|_{M_{i}}\to\mathcal{I}$. Thus $s|_{V_{i}}-s_{i}$ has image in
$I_{1}^{2}$ so using the same argument as in (2.23, 2.24),

(5.26) $s|_{V_{i}}\ =\ s_{i}+O\big{(}s_{i}^{2}\big{)}$

in the sense of Definition 2.3. Thus the $(\psi_{ij},\Phi_{ij})$ glue the
$s_{i}$, up to an $O(s_{i}^{2})$ correction, to give $s$. By Definition 2.4
this means $(\psi_{ij},\Phi_{ij})$ glue the holomorphic Kuranishi charts
$(V_{i},E_{i},s_{i})$.

Both $\psi_{ij}$ and $\Phi_{ij}$ satisfy the cocycle condition and so give us
the global ambient space $V$ and the global $\mathcal{A}_{1}$-bundle $E$ over
it. Then we have the global $\mathcal{A}_{1}$-section $s$ defining the global
Kuranishi chart $(V,E,s)$.

We end by noting that by Theorem 3.3 we can replace each of these
$\mathcal{A}$-gluings $(\psi_{ij},\Phi_{ij})$ by a homotopic

(5.27) $\text{ \emph{holomorphic} gluing
}\,\big{(}\psi^{\prime}_{ij},\,\Phi^{\prime}_{ij}\big{)}.$

The $\psi^{\prime}_{ij}$ (or $\Phi^{\prime}_{ij}$) do not generally satisfy
the cocycle condition on their own, so we do not get a holomorphic structure
on $V$ (or $E$). But as morphisms of complex Kuranishi charts, the pairs
(5.27) do satisfy the cocycle condition up to homotopy in the sense of
Definition 2.17. They holomorphically glue the holomorphic Kuranishi charts
$(V_{i},E_{i},s_{i})$ to define the same complex Kuranishi structure as the
global $\mathcal{A}$-Kuranishi chart $(V,E,s)$ of Theorem 2.

##  6\. Cones

Again let $(M,E^{\bullet})$ be a projective scheme with perfect obstruction
theory. From this data Beherend-Fantechi produce a cone $C\subset
E_{1}:=(E^{-1})^{*}$ by pulling back their intrinsic normal cone from the
stack $E_{1}/E_{0}$ (called $h^{1}/h^{0}\big{(}(E^{\bullet})^{\vee}\big{)}$ in
[BF]).

Their construction is algebraic, of course. We can describe it locally in
terms of the algebraic Kuranishi charts (5.24) made from the perfect
obstruction theory. Let $I$ denote the ideal of
$M_{i}\subset\widetilde{K}_{i}$, so that $I/I^{2}=\mathcal{I}$. Then the cone
is

(6.1) $\mathrm{Spec}\,\bigoplus_{n\geq 0}I^{n}/I^{n+1}\ \subset\
\mathrm{Spec}\,\bigoplus_{i\geq 0}\,\operatorname{Sym}^{n}(I/I^{2})\ =\
\mathrm{Spec}\,\bigoplus_{i\geq 0}\,\operatorname{Sym}^{n}\mathcal{I}\
\subset\ E_{1},$

This glues over overlaps $M_{ij}$, even though $I$ does not in general. Since
$\mathcal{I}$ and $E_{1}$ are global sheaves over $M$, the right hand side of
(6.1) obviously glues. It is less obvious, but also true, that the left hand
side glues to give the cone $C\subset E_{1}$ over $M$.

Denote the inclusion map from the total space of $E|_{M}$ to the total space
of $E$ over $V\subset\widetilde{K}^{*}$ (the global Kuranishi chart of Section
5.3) by

(6.2) $j\ \colon\,E_{1}\,\lhook\joinrel\relbar\joinrel\rightarrow\,E.$

Inside $E$ we will next show the Borel-Moore fundamental class $j_{*}[C]$ is
homologous to the graph $\Gamma_{\\!s}\subset E$ of $s$. For this we will need
some notation.

###  $\epsilon$-neighbourhoods

Fix a projective subscheme $Z$ of a quasi-projective scheme $X$. Pulling back
the Fubini-Study metric from a projective algebraic embedding
$X\hookrightarrow\mathbb{P}^{n}$ defines a metric $d$ on $X$. Then we let

(6.3) $U_{X}^{\epsilon}(Z)\ :=\ \big{\\{}x\in
X\,\colon\,d(x,Z)<\epsilon\big{\\}}$

with closure $\overline{U}_{\\!X}^{\hskip 0.6pt\epsilon}(Z)=\\{x\in X\colon
d(x,Z)\leq\epsilon\\}$ and boundary

$\partial\overline{U}_{\\!X}^{\hskip 0.6pt\epsilon}(Z)\ :=\ \big{\\{}x\in
X\,\colon\,d(x,Z)=\epsilon\big{\\}}.$

This is empty if $Z=X$. We will always assume — but often omit to say — that
$\epsilon>0$ is taken sufficiently small, and we permit ourselves to shrink it
during proofs. Thus the choice of $d$ is unimportant, and $Z\subset
U_{X}^{\epsilon}(Z)$ is a deformation retract. In particular the inclusion
induces an isomorphism $H_{*}(Z)\cong H_{*}\big{(}U_{X}^{\epsilon}(Z)\big{)}$.

######  Proposition 6.4.

In the total space of $E$ over $V\\!$, let $U=U_{E}^{\epsilon}(M)$. Then

(6.5) $\big{[}\Gamma_{\\!s}\cap\overline{U}\hskip 0.6pt\big{]}\ =\
\big{[}j_{*}\hskip 0.6ptC\cap\overline{U}\hskip 0.6pt\big{]}\ \in\
H_{*}\big{(}\,\overline{U},\,\partial\overline{U}\\!\smallsetminus
0_{E}\big{)}.$

Here $0_{E}$ is the zero section of $E$ and
$\partial\overline{U}\\!\smallsetminus 0_{E}$ is shorthand for
$\partial\overline{U}\\!\smallsetminus(\partial\overline{U}\cap 0_{E})$.

###### Proof.

Consider the union of the graphs

(6.6) $\bigcup_{t\in[1,\infty)}\Gamma_{\\!ts}\ \subset\ E.$

This naturally defines a Borel-Moore chain in $E$ with boundary
$\Gamma_{\\!s}$ at $t=1$. We claim its closure, intersected with
$\overline{U}$, defines a chain in
$C_{*}(\overline{U},\partial\overline{U}\\!\smallsetminus 0_{E})$ with
additional boundary the fundamental class of $j_{*}\hskip 0.6ptC$ at
$t=\infty$.

To prove this it is sufficient to work locally. Thus we may replace $(V,E)$ by
$\big{(}V_{i}:=V\cap\widetilde{K}_{i}^{*},E_{i}\big{)}$ (5.23),
$U=U_{E}^{\epsilon}(M)$ by $U_{i}:=U\cap E_{i}$ and $s$ by $s_{i}$ (5.24).
These are all algebraic, and we recall from (5.26) that

(6.7) $s|_{V_{i}}\ =\ s_{i}+O\big{(}s_{i}^{2}\big{)}.$

It is standard that, in the Hilbert scheme of algebraic subschemes of $E_{i}$,
there is a unique limit of the algebraic graphs $\Gamma_{\\!ts_{i}}$ and it is
the cone (6.1) pushed forward by $j$ (6.2). See for example [Fu, Remark
5.1.1.]. Therefore $j_{*}[C]$ is also the limit in the space of integral
currents.202020We thank Vivek Shende for suggesting we use integral currents.
Since they carry fundamental homology classes they are a good replacement in
the $C^{\infty}$ setting for the use of subschemes in the algebraic case. So
we are left with checking that the estimate (6.7) implies the graphs
$\Gamma_{\\!ts}$ and $\Gamma_{\\!ts_{i}}$ have the same limits _as integral
currents_ as $t\to\infty$.

We write $e_{i}:=s-s_{i}$. In any hermitian metric on $E_{i}$ we have an
estimate

$|e_{i}|\ \leq\ C_{1}|s_{i}|^{2}$

from (6.7). Thus on $\Gamma_{\\!ts_{i}}\cap U_{i}\subset E_{i}$ we have

(6.8) $|ts_{i}|\ <\ \epsilon\quad\text{and so}\quad|te_{i}|\ \leq\
\frac{C_{1}\epsilon^{2}}{|t|}\,.$

Fix any compactly supported $C^{\infty}$ test form $\sigma$ on $U_{i}$ and
apply Stokes’ theorem to the chain

$\Delta=\bigcup_{\lambda\in[0,t]}\Gamma_{\\!ts_{i}+\lambda e_{i}}\
\cap\,U_{i}$

bounding the two graphs. Using (6.8) to bound its thickness, and the fact that
$\sigma$ is zero on $\partial\overline{U}_{i}$, we get the estimate

(6.9) $\left|\int_{\Gamma_{\\!ts_{i}}\cap\hskip
0.6ptU_{i}}\sigma-\int_{\Gamma_{\\!ts\hskip 0.6pt}\cap\hskip
0.6ptU_{i}}\sigma\right|\ \leq\
\left|\int_{\Delta}d\sigma\right|+\left|\int_{\overline{\Delta}\cap\hskip
0.6pt\partial\overline{U}_{i}}\sigma\right|\,\stackrel{{\scriptstyle\eqref{esti}}}{{\leq}}\,\frac{C_{2}\epsilon^{2}}{|t|}\,.$

As $t\to\infty$, (6.9) tends to zero, so

$\lim_{t\to\infty}\int_{\Gamma_{\\!ts}\hskip 0.6pt\cap\hskip
0.6ptU_{i}}\sigma\ =\ \lim_{t\to\infty}\int_{\Gamma_{\\!ts_{i}}\hskip
0.6pt\cap\hskip 0.6pt\overline{U}_{i}}\sigma\ =\ \int_{C\hskip 0.6pt\cap\hskip
0.6pt\overline{U}_{i}}\sigma.$

Thus the integral currents $\Gamma_{\\!ts}$ tend to $j_{*}\hskip 0.6ptC$ on
$U_{i}$, as required. Finally we note that the graphs and $\Delta$ intersect
$0_{E}$ (in the set-theoretic sense) only in $M$, so they all miss
$\partial\overline{U}\cap 0_{E}$. ∎

### Virtual cycles

For a general Joycian $\mu$-Kuranishi structure there is also a global
Kuranishi chart [Jo2, Section 14 and Corollary 4.36]. In that setting the
virtual cycle is defined topologically in the deformation retract
$\overline{U}$ of $M$ by taking the homology class

(6.10) $\big{[}\Gamma_{\\!s}\cap\overline{U}\hskip 0.6pt\big{]}\ \in\
H_{\dim_{\mathbb{R}\\!}V}\big{(}\,\overline{U},\,\partial\overline{U}\\!\smallsetminus
0_{E}\big{)}$

and capping it with the Thom form

$\big{[}0_{E}\big{]}\ \in\ H^{\hskip
0.6pt\operatorname{rank}_{\mathbb{R}\\!}E}\big{(}\,\overline{U},\,\partial\overline{U}\\!\smallsetminus
0_{E}\big{)}.$

This defines the virtual cycle as their topological intersection

(6.11) $\Gamma_{\\!s}\cdot 0_{E}\ \in\ H_{i}\big{(}\hskip
0.6pt\overline{U},\mathbb{Z}\big{)}\ \cong\ H_{i}(M,\mathbb{Z}),\qquad
i\,=\,\dim_{\mathbb{R}\\!}V-\operatorname{rank}_{\mathbb{R}\\!}E.$

When $M$ is projective with perfect obstruction theory the graphs
$\Gamma_{\\!ts}$ limit to the cone $C$ as $t\to\infty$. This has the advantage
of lying in $E_{1}=E|_{M}$ over $M$, instead of just being inside $E$ over
$V$. Now $i=2v$ where $v$ is the virtual dimension
$\dim_{\mathbb{C}\\!}V-\operatorname{rank}_{\mathbb{C}\\!}E$ and the Behrend-
Fantechi virtual cycle is

(6.12) $0_{E_{1}}^{\hskip 0.6pt!}[C]\ \in\ A_{v}(M)\longrightarrow
H_{2v}(M,\mathbb{Z}).$

######  Proposition 6.13.

The two virtual cycles _( 6.11, 6.12)_ agree.

###### Proof.

Proposition 6.4 equates (6.10) with $[j_{*}\hskip 0.6ptC\cap\overline{U}\hskip
0.6pt]\in H_{\dim
V}\big{(}\,\overline{U},\,\partial\overline{U}\\!\smallsetminus 0_{E}\big{)}$
by (6.5). Thus (6.11) equals the cap product

$H_{2v}\big{(}\hskip 0.6pt\overline{U},\mathbb{Z}\big{)}\ \ni\ j_{*}\hskip
0.6ptC\cdot 0_{E}\ =\ j_{*}(C\cdot j^{*}0_{E})\ =\ j_{*}(C\cdot 0_{E_{1}})\ =\
j_{*}(0_{E_{1}}^{\hskip 0.6pt!}[C]).\qed$

See [LT2, Sie] for an analogous result in the more difficult case of $M$ being
a Deligne-Mumford stack. In fact Siebert [Sie, Section 3] finds a way to take
a limit of the graphs $\Gamma_{\\!ts}$ to define a generalised cone even in
non-holomorphic settings; this recovers the Behrend-Fantechi cone when
everything is algebraic.

##  7\. Moduli of sheaves on Calabi-Yau fourfolds

Unfortunately Proposition 6.13 is not relevant for the topic of the rest of
this paper — namely the virtual cycles of [BJ, OT] on moduli spaces $M$ of
stable sheaves on Calabi-Yau 4-folds. These are defined by intersecting
$\Gamma_{s}$ (or $C$) _not_ with the zero section of $E$ (or $E|_{M}=E_{1}$)
but _with a half dimensional subbundle_ instead.

Before describing the theory, we give a brief overview of the plan in the case
that the virtual dimension $\operatorname{vd}$ (0.8) is even.

We will see we can take $E_{1}$ to be an $SO(2n,\mathbb{C})$ bundle inside
which $C$ is isotropic. Moreover $E$ and $s$ are made from patching together
$SO(2n,\mathbb{C})$ bundles and isotropic sections. Then

  1. (i)

the Borisov-Joyce virtual cycle [BJ] equals the intersection of
$\Gamma_{\\!s}$ with a maximal negative definite real subbundle
$iE^{\mathbb{R}}\subset E$, while

  2. (ii)

the algebraic virtual cycle [OT] is $(-1)^{n}$ times by the intersection of
$C$ with a positive212121In the sense of [OT, Definition 2.2]. maximal
isotropic subbundle $\Lambda\subset E_{1}=E|_{M}$.

We prove (i) in Theorem 8.1; here the intersection lies naturally in the zero
section $V$ of $E$. In (ii) we have to use cosection localisation [KL] to
force the intersection into the zero section $M$ of $E|_{M}$.

We will prove the two cycles are equal in two main steps,

  1. (1)

homotoping $\Gamma_{\\!s}$ to $C$ as in Proposition 6.4 (Theorem 7.23) _and_

  2. (2)

homotoping $E^{\mathbb{R}}\subset E$ to $\Lambda\subset E$ (Theorem 10.2).

Let $(X,\mathcal{O}_{X}(1))$ be a smooth polarised Calabi-Yau 4-fold:
$K_{X}\cong\mathcal{O}_{X}$. Let $M=M_{c}(X)$ be the moduli space of
$\mathcal{O}_{X}(1)$-Gieseker stable sheaves $F$ on $X$ of a fixed Chern
character $c$, selected so that all semistable sheaves with Chern character
$c$ are stable. Thus $M$ is projective.

We recall the deformation theory of $M$ from [OT]. Given a universal twisted
sheaf $\mathcal{F}$ on $\pi:X\times M\to M$, we let

(7.1) $\mathbb{E}\ :=\ \tau^{[-2,0]}\big{(}R\pi_{*}R\curly
H\\!om(\mathcal{F},\mathcal{F})[3]\big{)}.$

Relative Serre duality down the map $\pi$ gives an isomorphism

(7.2) $\theta\,\colon\,\mathbb{E}\xrightarrow{\ \sim\
}\mathbb{E}^{\vee}[2]\quad\text{such that}\quad\theta\ =\ \theta^{\vee}[2]\
\in\ \operatorname{Ext}^{2}(\mathbb{E},\mathbb{E}^{\vee}).$

Using the Atiyah class of $\mathcal{F}$ as in [HT] defines a morphism

(7.3) $\displaystyle\phi\,\colon\,\mathbb{E}\longrightarrow\mathbb{L}_{M}$

which, by [HT, Theorem 4.1], is an obstruction theory in the sense of [BF,
Definition 4.4]. In other words, $h^{0}(\phi)\colon
h^{0}(\mathbb{E})\to\Omega_{M}$ is an isomorphism, and
$h^{-1}(\phi):h^{-1}(\mathbb{E})\to h^{-1}(\mathbb{L}_{M})$ is a surjection.
Its virtual dimension is (0.8),

(7.4) $\operatorname{vd}\ :=\ \operatorname{rank}\mathbb{E}\ =\ 2-\chi(c,c).$

However $\phi$ is _not a perfect obstruction theory_ because $\mathbb{E}$ is
perfect of amplitude $[-2,0]$ rather $[-1,0]$. (Equivalently, in general
$\operatorname{Ext}^{3}(F,F)=\operatorname{Ext}^{1}(F,F)^{*}$ is nonzero for
$F\in M$, so $\mathbb{E}$ can have $h^{-2}\neq 0$.) So we cannot apply [BF] to
get a virtual cycle.

To describe the virtual cycles of [BJ, OT] we use a standard form for the
obstruction theory (7.3).

######  Proposition 7.5.

[OT, Propositions 4.1 and 4.2] Fix an embedding $M\subset\mathbb{P}^{N}$ with
ideal $J\subset\mathcal{O}_{\mathbb{P}^{N}}$. On $M$ there is a bundle $T$, an
$SO(r,\mathbb{C})$ bundle $E_{1}$, and a surjective map of complexes

(7.6)
---
$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{*}}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\
T^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\bullet}\\!\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{=}$$\textstyle{J/J^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{\Omega_{\mathbb{P}^{N}}\big{|}_{M}}$$\textstyle{\mathbb{L}_{M}\\!}$

which represents the obstruction theory $\mathbb{E}\to\mathbb{L}_{M}$ (7.3).

Here the quadratic form $q$ on $E_{1}$ gives the isomorphism $q\colon
E_{1}\cong E_{1}^{*}$ used in defining $a^{*}$. The quasi-isomorphism
$\mathbb{E}\cong E^{\bullet}$ intertwines the self-dualities

$(\operatorname{id}_{T},\,q,\,\operatorname{id}_{T^{*}})\
\colon\,E^{\bullet}\xrightarrow{\ \sim\
}E_{\bullet}[2]\quad\text{and}\quad\theta\
\colon\,\mathbb{E}\xrightarrow[\eqref{duel}]{\ \sim\
}\mathbb{E}^{\vee}[2].\vspace{-9mm}$

$\hfill\square$

Consider the stupid truncation $\tau E^{\bullet}$ of the obstruction theory
(7.6),

(7.7)
$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{T^{*}\\!\\!\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J/J^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{\Omega_{\mathbb{P}^{N}}\big{|}_{M}.}$

As in [OT, Section 4.2] this defines a stupid perfect obstruction
theory222222This perfect obstruction theory ignores the degree $-2$ term $T$
in $E^{\bullet}$, so has the wrong virtual dimension and is not quasi-
isomorphism invariant. The papers [BJ, OT] effectively replace $E_{1}$ by
different half-dimensional subbundles when discarding $T$.  for $M$ of virtual
dimension

$v\ =\ \operatorname{rank}T-\operatorname{rank}E_{1}\ =\
\operatorname{vd}-\operatorname{rank}T,$

cf. (4.2). By [BF] this defines a cone

(7.8) $C\ \subset\ E_{1}$

over $M$ by pulling back the intrinsic normal cone from
$h^{1}/h^{0}\big{(}(\tau E^{\bullet})^{\vee}\big{)}=[E_{1}/T]$. Considering
$q$ as a function on the total space of $E_{1}$ (quadratic on each fibre),
[OT, Proposition 4.3] shows it restricts to 0 on $C$.

######  Proposition 7.9.

The cone $C\subset E_{1}$ is isotropic: $q|_{C}\equiv 0.\hfill\square$

Using the stupid perfect obstruction theory (7.7), Section 5 now gives us

(7.10) $\text{a global complex Kuranishi chart }\,(V,E,s)\,\text{ for }M.$

Recall the ambient space $V$ is a neighbourhood of $M$ inside the total space
of $\widetilde{K}^{*}$ — an $\mathcal{A}_{1}$-bundle over
$\mathring{\mathbb{P}}^{N}$ extending the dual of
$K:=\ker\big{(}T^{*}\to\to\Omega_{\mathbb{P}^{N}}|_{M}\big{)}$ over $M$. Its
cotangent sheaf satisfies $\Omega_{V}|_{M}\cong T^{*}$. Similarly $E$ is an
$\mathcal{A}_{1}$-bundle over $V$ extending $E_{1}$ over $M$, and $s$ is an
$\mathcal{A}_{1}$-section of $E$ whose restriction to $2M\subset V$ has
derivative the map $a$ in (7.7). The induced diagram (5.25),

$\textstyle{E^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds}$$\scriptstyle{s}$$\textstyle{\Omega_{V}\big{|}_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}/I_{1}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{V}\big{|}_{M}\hskip
0.6pt.\\!\\!}$

is precisely (7.7). Therefore

(7.11) $T_{V}|_{M}\xrightarrow{\ (ds)^{*}\ }E^{*}|_{M}\xrightarrow{\ ds\
}\Omega_{V}|_{M}$

is the top row $E^{\bullet}$ of (7.6); in particular it is a complex.

### Real virtual cycle

Since $SO(r,\mathbb{C})$ is homotopic to $SO(r,\mathbb{R})$ there is a real
bundle $E_{1}^{\mathbb{R}}$ underlying $E_{1}$ in the sense that

(7.12) $E_{1}\ =\ E_{1}^{\mathbb{R}}\oplus iE_{1}^{\mathbb{R}}$

with $E_{1}^{\mathbb{R}}\subset E_{1}$ a maximal
($\operatorname{rank}_{\mathbb{R}}E_{1}^{\mathbb{R}}=r$) positive definite
real subbundle for $q$. In particular $q$ is real and _negative definite_ on
$iE_{1}^{\mathbb{R}}$.

By contrast $q$ _vanishes_ on the isotropic cone $C\subset E_{1}$, so $C$ and
$iE_{1}^{\mathbb{R}}$ can only intersect in the 0-section $M$. Thus we may
intersect $C$ and $iE_{1}^{\mathbb{R}}$ inside $E_{1}$ by taking the homology
class

$[C]\ \in\ H_{2\operatorname{rank}T}\Big{(}\hskip 0.6pt\overline{U}^{\hskip
0.6pt\epsilon}_{\\!E_{1}}(M),\ \partial\overline{U}^{\hskip
0.6pt\epsilon}_{\\!E_{1}}(M)\\!\smallsetminus iE_{1}^{\mathbb{R}}\Big{)},$

and capping with the Thom form

(7.13) $\big{[}iE_{1}^{\mathbb{R}}\big{]}\ \in\ H^{r}\Big{(}\hskip
0.6pt\overline{U}^{\hskip 0.6pt\epsilon}_{\\!E_{1}}(M),\
\partial\overline{U}^{\hskip 0.6pt\epsilon}_{\\!E_{1}}(M)\\!\smallsetminus
iE_{1}^{\mathbb{R}}\Big{)}.$

Here the sign of the Thom form of $iE_{1}^{\mathbb{R}}$ is fixed by a choice
of orientation on the fibres of its normal bundle (the pullback of)
$E_{1}^{\mathbb{R}}$. But (real) orientations on $E_{1}^{\mathbb{R}}$ are
equivalent to complex orientations on the $SO(r,\mathbb{C})$ bundle $E_{1}$,
as described carefully in [OT, Section 2.2]. And by [OT, Proposition 4.2]
complex orientations on $E_{1}$ are equivalent to complex orientations on
$\mathbb{E}$ (7.1). A choice of the latter is guaranteed by [CGJ] and fixed
once and for all throughout the papers [BJ, OT] to define their virtual
cycles.232323The definitions of complex orientations on orthogonal vector
spaces and bundles ([BJ, Equation 2.7], [OT, Definition 2.1]) and on self-dual
complexes ([BJ, Definition 2.12] and [OT, Equation 59] use different sign
conventions but are equivalent.

######  Definition 7.14.

By [CGJ] we may fix a complex orientation on $\mathbb{E}$. As above this
orients $E_{1}^{\mathbb{R}}$ and hence defines a Thom class (7.13). We then
define the real virtual class to be

(7.15) $C\cdot iE_{1}^{\mathbb{R}}\ \in\ H_{\operatorname{vd}}(M,\mathbb{Z}).$

To relate (7.15) to the Borisov-Joyce class [BJ] (from which it will follow it
is independent of choices) we show how to replace $C$ by $\Gamma_{\\!s}$. For
this we extend the splitting (7.12) over the neighbourhood $V$ of
$M\subset\widetilde{K}^{*}$,

(7.16) $E\ =\ E^{\mathbb{R}}\oplus iE^{\mathbb{R}},$

and extend the holomorphic quadratic form $q$ on $E_{1}$ to a $C^{\infty}$
quadratic form $q$ on $E$. Shrinking $V$ if necessary we may assume that Re
$q$ is positive definite on $E^{\mathbb{R}}$. Together the quadratic form and
the real structure (7.16) define a hermitian metric $|\,\cdot\,|$ on $E$.

Pushing $C$ forward from $E_{1}=E|_{M}$ to $E$ via $j$ (6.2) and setting
$U:=U^{\hskip 0.6pt\epsilon}_{\\!E}(M)$ as before we can rewrite (7.15) by
taking the homology class

$[j_{*}\hskip 0.6ptC]\ \in\ H_{\dim_{\mathbb{R}}\\!V}\big{(}\hskip
0.6pt\overline{U},\,\partial\overline{U}\\!\smallsetminus
iE^{\mathbb{R}}\big{)},$

and capping with the Thom form

(7.17) $\big{[}iE^{\mathbb{R}}\big{]}\ \in\
H^{\operatorname{rank}E}\big{(}\overline{U},\,\partial\overline{U}\\!\smallsetminus
iE^{\mathbb{R}}\big{)}$

to give

(7.18) $j_{*}\hskip 0.6ptC\cdot iE^{\mathbb{R}}\ \in\
H_{\operatorname{vd}}\big{(}\hskip 0.6pt\overline{U},\mathbb{Z}\big{)}\ \cong\
H_{\operatorname{vd}}(M,\mathbb{Z}).$

But $j^{*}[iE^{\mathbb{R}}]=[iE_{1}^{\mathbb{R}}]$ so (7.15) and (7.18)
coincide,

(7.19) $C\cdot iE_{1}^{\mathbb{R}}\ =\ j_{*}\hskip 0.6ptC\cdot
iE^{\mathbb{R}}.$

Next we claim that for $0<\epsilon\ll 1$ sufficiently small, $\Gamma_{\\!s}$
also avoids242424In fact the set-theoretic intersection of $\Gamma_{\\!s}$ and
$iE^{\mathbb{R}}$ is precisely $M$. With some work this follows from analysing
the gluing in Appendix A.3 applied to the local Darboux models $(E_{i},s_{i})$
of [BBJ] and [OT, Equation 68] consisting of isotropic sections $s_{i}$ of
orthogonal bundles $E_{i}$ over the open cover $V_{i}$ of $V$.
$\partial\overline{U}\cap iE^{\mathbb{R}}$. Since $\Gamma_{\\!ts}$ limits to
the isotropic cone $C$ as $t\to\infty$ we can replace $s$ by $ts$ for $t$
sufficiently large (or equivalently, by the $\mathbb{C}^{*}$-invariance of
$C$, take $\epsilon$ sufficiently small) to ensure that on
$\Gamma_{\\!s}\cap\partial\overline{U}$,

  * •

$q$ is as small as we like, and

  * •

$|s|^{2}$ is as close as we like to $\epsilon$.

So we can easily arrange for the following to hold,

$|q(s,s)|\ <\ |s|^{2}\,\text{ on }\,\Gamma_{\\!s}\cap\partial\overline{U}.$

Since the estimate only improves as we scale $s$ by $t>1$ we note for later
the stronger inequality

(7.20) $|q(e,e)|\ <\ |e|^{2}\,\text{ for
}\,e\,\in\,\Gamma_{\\!ts}\cap\partial\overline{U},\ \ \forall t\,\geq\,1.$

Conversely, on $iE^{\mathbb{R}}$, by the definition of the Hermitian metric
$|\,\cdot\,|$,

(7.21) $q(ie,ie)\ =\ -|e|^{2}\,\text{ for
}\,ie\,\in\,iE^{\mathbb{R}}\cap\partial\overline{U}.$

Therefore $\Gamma_{\\!s}$ and $iE^{\mathbb{R}}$ do not meet in
$\partial\overline{U}$. So we may cap

$[\Gamma_{\\!s}]\ \in\ H_{\dim_{\mathbb{R}}\\!V}\big{(}\hskip
0.6pt\overline{U},\,\partial\overline{U}\\!\smallsetminus
iE^{\mathbb{R}}\big{)}$

with the Thom form $\big{[}iE^{\mathbb{R}}\big{]}$ (7.17) to define

(7.22) $\Gamma_{\\!s}\cdot iE^{\mathbb{R}}\ \in\
H_{\operatorname{vd}}\big{(}\hskip 0.6pt\overline{U},\mathbb{Z}\big{)}\ \cong\
H_{\operatorname{vd}}(M,\mathbb{Z}).$

Then the analogue of Proposition 6.4 for this intersection is the following.

######  Theorem 7.23.

Under the above conditions (7.15) and (7.22) coincide,

(7.24) $\Gamma_{\\!s}\cdot iE^{\mathbb{R}}\ =\ C\cdot iE_{1}^{\mathbb{R}}.$

###### Proof.

The proof of Proposition 6.4 showed the chain (6.6),

$\Gamma\ :=\ \bigcup\nolimits_{t\in[1,\infty)}\Gamma_{\\!ts},$

bounds $[\Gamma_{\\!s}]-j_{*}[C]$. Comparing (7.20) with (7.21) shows $\Gamma$
is disjoint from $iE^{\mathbb{R}}\cap\partial\overline{U}$. Thus we get the
following strengthening of (6.5),

$[\Gamma_{s}]\ =\ j_{*}[C]\ \in\ H_{\dim
V}\big{(}\overline{U},\,\partial\overline{U}\\!\smallsetminus
iE^{\mathbb{R}}\big{)}.$

Capping with (7.17) now gives $\Gamma_{\\!s}\cdot iE^{\mathbb{R}}=j_{*}\hskip
0.6ptC\cdot iE^{\mathbb{R}}$, which by (7.19) gives $C\cdot
iE_{1}^{\mathbb{R}}$ as required. ∎

By the projection formula the left hand side of (7.24) can be expressed as an
intersection in $E^{\mathbb{R}}$. Letting $s^{+}$ denote the projection of $s$
to $E^{\mathbb{R}}$ it gives

(7.25) $\Gamma_{\\!s^{+}}\cdot 0_{E^{\mathbb{R}}}\ =\ C\cdot
iE^{\mathbb{R}}_{1}.$

The left hand side is the virtual cycle associated to the global
$\mu$-Kuranishi chart

(7.26) $\big{(}V,E_{1}^{\mathbb{R}},s^{+}\big{)}.$

The next Section is devoted to proving that this is a $\mu$-Kuranishi chart
for $M$ compatible with (one of) Borisov-Joyce’s $\mu$-Kuranishi structure on
$M$. While their $\mu$-Kuranishi structures depend on choices, the cobordism
(or homology) class of the resulting virtual cycle
$[M]_{BJ}^{\operatorname{vir}}$ is uniquely defined by [BJ, Corollary 3.19].

##  8\. The Borisov-Joyce virtual cycle

In this Section we prove the following result.

######  Theorem 8.1.

The global complex Kuranishi chart (7.26) induces one of the Borisov-Joyce
$\mu$-Kuranishi structures of [BJ, Theorem 3.15] on $M$. Thus, by (7.25),

$[M]^{\operatorname{vir}}_{BJ}\ =\ \Gamma_{\\!s^{+}}\cdot 0_{E^{\mathbb{R}}}\
=\ C\cdot iE^{\mathbb{R}}_{1}.$

###  8.1. Review of Borisov-Joyce’s construction

We begin by describing Borisov-Joyce’s $\mu$-Kuranishi structure on $M$. They
rely heavily on the canonical $(-2)$-shifted symplectic derived scheme
structure on $M$. The derived part of this structure is defined in [TVa] as a
refinement252525Strictly speaking [TVa] describes a derived moduli _stack_ of
sheaves on $X$. We rigidify by removing the $\mathbb{C}^{*}$ stabilisers from
the substack of stable sheaves to recover $M$. At the same time we remove
$h^{1}\cong\mathrm{Lie}(\mathbb{C}^{*})^{*}\otimes\mathcal{O}$ from the
derived cotangent bundle, and the dual copy of $\mathcal{O}$ from $h^{-3}$, to
get the $(-2)$-shifted symplectic derived scheme structure on $M$. of the
obstruction theory (7.3). The $(-2)$-shifted symplectic structure is defined
in [PTVV] as a refinement of the Serre duality quadratic structure (7.2).

By [BG, BBJ, BBBJ] this structure is equivalent to $M$ having

(8.2) a Zariski open cover $\\{M_{i}\\}$ by local Darboux charts
$(D_{i},F_{i},s_{i})$

with a certain compatibility between them which we will describe in (8.5)
below. Here

  * •

$D_{i}$ is a smooth scheme,

  * •

$F_{i}$ is a holomorphic _orthogonal_ bundle over it,

  * •

$s_{i}$ a holomorphic _isotropic_ section of $F_{i}$,

and we fix an isomorphism (which we are supressing in the notation) between
the zero scheme of $s_{i}$ and $M_{i}$.

It is important that the Darboux charts (8.2) should not really be thought of
as Kuranishi charts — this would _not_ give the Borisov-Joyce Kuranishi
structure262626Analogously in [OT] we do not take the Behrend-Fantechi virtual
cycle (6.12) that comes from considering $(V,E,s)$ as a global Kuranishi
chart; instead we take (10.1). on $M$. In fact thinking of them as Kuranishi
charts would correspond to using the Koszul cdga

(8.3) $(\Lambda^{\bullet}F_{i}^{*},s_{i})\ =\
\cdots\longrightarrow\Lambda^{2}F_{i}^{*}\xrightarrow{\ s_{i}\
}F_{i}^{*}\xrightarrow{\ s_{i}\ }\mathcal{O}_{D_{i}}$

to describe the derived structure on $M_{i}$, with virtual cotangent bundle

$F_{i}^{*}|_{M_{i}}\xrightarrow{\ ds_{i}\ }T^{*}_{D_{i}}|_{M_{i}}.$

But this is the stupid truncation of the actual virtual cotangent bundle

$T_{D_{i}}|_{M_{i}}\xrightarrow{\ ds_{i}\ }F_{i}^{*}|_{M_{i}}\xrightarrow{\
ds_{i}\ }T^{*}_{D_{i}}|_{M_{i}},$

which is the virtual cotangent bundle of the full cdga

(8.4)
$\Lambda^{\bullet}F_{i}^{*}\otimes\operatorname{Sym}^{\bullet}\\!\big{(}T_{D_{i}}[2]\big{)}\
=\ \\\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
20.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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20.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Lambda^{3}F_{i}^{*}\oplus(T_{D_{i}}\\!\otimes\\!\hskip
0.6ptF_{i}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
107.702pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-4.0pt\hbox{$\scriptstyle{\hskip
0.6pt\tiny\Big{(}\\!\\!\\!\begin{array}[]{cccc}s_{i}\\!\\!&\\!\\!\nabla
s_{i}\\\ 0\\!\\!&\\!\\!s_{i}\end{array}\\!\\!\\!\\!\Big{)}\hskip
0.6pt}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
150.38553pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
110.38553pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Lambda^{2}F_{i}^{*}\oplus
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197.63628pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
213.00565pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 233.00565pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{F_{i}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
250.42471pt\raise 5.16647pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 262.75436pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
262.75436pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{O}_{D_{i}}.}$}}}}}}}\ignorespaces}}}}\ignorespaces$

Here $\nabla$ is any holomorphic connection on $F_{i}$ with respect to which
its quadratic form is parallel.272727Beware we omit the differentials from the
shorthand notation
$\Lambda^{\bullet}F_{i}^{*}\otimes\operatorname{Sym}^{\bullet}\\!\big{(}T_{D_{i}}[2]\big{)}$.
This is the correct cdga to assign to the Darboux chart (8.2), and any cdga
(8.4) made from a Darboux chart in this way carries a canonical $(-2)$-shifted
symplectic derived structure. This describes the $(-2)$-shifted symplectic
structure on $M_{i}$ which comes from restricting the $(-2)$-shifted
symplectic structure of [PTVV] on $M$. Stupidly truncating by removing the
$T_{D_{i}}$ terms recovers the Koszul cdga (8.3).

The compatibility between (or gluing of) the Darboux charts (8.2) is given by
homotopy equivalences of the cdgas (8.4). Given a cover by Darboux charts, the
$(-2)$-shifted symplectic derived structure on $M_{i}$ is equivalent to the
one (8.4) given by its Darboux chart (8.2); composing this equivalence with
the inverse of the corresponding equivalence from the Darboux chart
$(D_{j},F_{j},s_{j})$ defines the transition map of cdgas

(8.5)
$\Lambda^{\bullet}F_{i}^{*}\otimes\operatorname{Sym}^{\bullet}\\!\big{(}T_{D_{i}}[2]\big{)}\Big{|}_{D_{ij}}\,\xrightarrow{\
a_{ij}\ }\
\Lambda^{\bullet}F_{j}^{*}\otimes\operatorname{Sym}^{\bullet}\\!\big{(}T_{D_{j}}[2]\big{)}\Big{|}_{D_{ij}}.$

In general this will _not_ give an equivalence or gluing of the truncated
cdgas (8.3), just some kind of gluing-modulo-lower-order-terms, as described
carefully in [BJ, Corollary 3.5].

Borisov-Joyce then “halve” the derived structure on $M$ by

  1. (1)

discarding the terms in degrees $\leq-2$ — i.e. truncating (8.4) to (8.3), or
considering the Darboux chart (8.2) as a complex Kuranishi chart,

  2. (2)

replacing each $F_{i}\cong F_{i}^{\mathbb{R}}\oplus iF_{i}^{\mathbb{R}}$ by a
maximal real positive definite subbundle $F_{i}^{\mathbb{R}}$, and

  3. (3)

projecting the $s_{i}\in\Gamma(F_{i})$ to
$s_{i}^{+}\in\Gamma(F_{i}^{\mathbb{R}})$.

In [BJ, Corollary 3.11 and Theorem 3.15(vi)] they show the
$F_{i}^{\mathbb{R}}$ may be chosen so that the gluing-modulo-lower-degree-
terms of [BJ, Corollary 3.5] projects to induce a genuine gluing of the
resulting new $\mu$-Kuranishi charts $(D_{i},F^{\mathbb{R}}_{i},s^{+}_{i})$.
(This is a gluing in the sense of Definition 4.3, but using $C^{\infty}$ maps
and functions.) The resulting $\mu$-Kuranishi structure depends on various
choices (such as the initial choice of local Darboux charts) but the induced
virtual cycle is independent up to cobordism.

Our approach will be to first show that our local holomorphic Kuranishi charts
$(V_{i},E_{i},s_{i})$ of (5.24) can be upgraded to Darboux charts as in (8.2).
Yet we will also show that by their very construction they can be made to glue
(not just up to lower order terms) as in Definition 4.3 to give the global
non-holomorphic chart $(V,E,s)$ constructed in Section 5.3. Thus in this case
the Borisov-Joyce gluing procedure described above will give what one might
expect: step (1) will recover the global chart $(V,E,s)$, then steps (2) and
(3) project it to the global chart $(V,E^{\mathbb{R}},s^{+})$ of (7.26).

### Our Darboux charts

First we will show that the local holomorphic Kuranishi charts
$(V_{i},E_{i},s_{i})$ of (5.24) can be made into Darboux charts.

######  Proposition 8.6.

Refining the cover $M=\cup_{i}M_{i}$ if necessary, we may choose an orthogonal
structure on $E_{i}$ so that $s_{i}$ is an isotropic section and
$(V_{i},E_{i},s_{i})$ is a Darboux chart for $M_{i}\subset M$.

###### Proof.

Fix a point $x\in M$ and $i$ such that $x\in M_{i}$. We will allow ourselves
to shrink the neighbourhood $M_{i}$ of $x$ throughout the proof. Repeating the
proof for any $x\in M$ and using compactness will give the required refinement
of $\cup_{i}M_{i}$.

Let $F$ be the sheaf on $X$ corresponding to $x\in M$, and let $q$ be the
quadratic form on $\operatorname{Ext}^{2}(F,F)$ given by Serre duality. By
[BBJ, Example 5.16] there is an open neighbourhood $A$ of
$0\in\operatorname{Ext}^{1}(F,F)$, and a map

(8.7) $s\,\colon A\longrightarrow\operatorname{Ext}^{2}(F,F)\,\text{ with
}\,ds|_{0}\,=\,0,\ q(s,s)\,=\,0\,\text{ and }\,Z(s)\,\cong\,M_{i},$

such that the obstruction theory on $M_{i}$ is quasi-isomorphic to

---
$\textstyle{\hskip 22.76219ptQ^{\bullet}\ \ :=\ \
\big{\\{}T_{A}|_{M_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{(ds)^{*}}$$\textstyle{\Omega_{A}|_{M_{i}}\big{\\}}\\!\\!\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}|_{M_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cong}$$\textstyle{I/I^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{\Omega_{A}|_{M_{i}}}$$\textstyle{\mathbb{L}_{\hskip
0.6ptM_{i}}.\\!\\!}$

Here $I$ is the ideal sheaf of $M_{i}\subset A$ and $Q$ is the trivial
orthogonal bundle with fibre $\operatorname{Ext}^{2}(F,F)$. Its quadratic form
gives an isomorphism $Q\cong Q^{*}$ which defines $(ds)^{*}$ and the self-
duality $Q^{\bullet}\cong Q_{\bullet}[2]$ which is intertwined with the Serre
duality of (7.2) by the quasi-isomorphism
$Q^{\bullet}\cong\mathbb{E}|_{M_{i}}$.

Furthermore, in [OT, Equation 70] we showed that the self-dual complex
$E^{\bullet}=\big{\\{}T_{V}|_{M}\to E_{1}\to\Omega_{V}|_{M}\big{\\}}$ of
(7.11) can be written — on restriction to $M_{i}$ — as the orthogonal direct
sum of $Q^{\bullet}$ and a self-dual acyclic complex $K^{\bullet}=\\{K^{0}\to
K^{1}\to K_{0}\\}$

$E^{\bullet}\ \cong\ Q^{\bullet}\oplus K^{\bullet}.$

Thus the self-duality $E^{\bullet}\cong E_{\bullet}[2]$ is the direct sum of
the self-dualities $Q^{\bullet}\cong Q_{\bullet}[2]$ and $K^{\bullet}\cong
K_{\bullet}[2]$.

Choosing an isotropic splitting of $K^{1}$ as in [OT, Footnote 7] we may write
$K^{\bullet}$ as $\\{K^{0}\to K^{0}\oplus K_{0}\to K_{0}\\}$ with the maps and
orthogonal structure the obvious ones. Shrinking $M_{i}$ if necessary we may
assume all our bundles are trivial, so that

$K^{\bullet}\ \cong\ \big{\\{}\mathcal{O}_{M_{i}}^{\oplus k}\xrightarrow{\
(1,0)\ }\mathcal{O}_{M_{i}}^{\oplus k}\oplus\mathcal{O}_{M_{i}}^{\oplus
k}\xrightarrow{\ (0,1)\ }\mathcal{O}_{M_{i}}^{\oplus k}\big{\\}}$

is the (pullback to $M_{i}$ of the) virtual cotangent bundle associated to the
following Darboux chart for the origin in $\mathbb{C}^{k}$,

(8.8) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.15279pt\hbox{$\scriptstyle{t}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 92.42464pt\raise-6.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{\hbox{}}\ignorespaces}}}}\ignorespaces\qquad\begin{array}[]{c}\\\
\\\ \\\ q(\underline{x},\underline{y})\ =\ \sum x_{i}y_{i},\\\
t(\underline{z})\ =\ (0,\underline{z}),\\\ q(t,t)=0.\end{array}$

Thus $E^{\bullet}|_{M_{i}}\to\mathbb{L}_{M_{i}}$ is isomorphic to the virtual
cotangent bundle associated to the product Darboux chart
(8.7)$\,\times\,$(8.8). In particular the perfect obstruction theory given by
its stupid truncation $\tau
E^{\bullet}|_{M_{i}}:=\big{\\{}E_{1}\to\Omega_{V}|_{M}\big{\\}}\to\mathbb{L}_{M}$
is isomorphic to the perfect obstruction theory induced by the product
_Kuranishi_ (rather than Darboux) chart (8.7)$\,\times\,$(8.8).

Thus Theorem 3.3 gives an isomorphism between the Kuranishi charts
$(V_{i},E_{i},s_{i})$ and (8.7)$\,\times\,$(8.8), allowing us to transfer the
Darboux structure over from the latter to the former. ∎

###  8.2. A homotopy

So now the $(-2)$-shifted symplectic derived structure on $M$ defines cdga
gluing maps $a_{ij}$ between the cdgas (8.5) induced by these local Darboux
charts. All that we really know about $a_{ij}$ is that its induced action from
the virtual cotangent bundle of $M_{i}$ to that of $M_{j}$ is the gluing map
which gives the virtual cotangent bundle of $M$. By (7.11) this is

(8.9) $T_{V}|_{M}\xrightarrow{\ ds\ }E\xrightarrow{\ ds\ }T^{*}_{V}|_{M},$

where $(V,E,s)$ is the global complex Kuranishi chart (7.10).

Therefore the gluing equivalence induced by $a_{ij}$ between the horizontal
3-term complexes below is equal, in $D^{b}(\mathrm{Coh}\,M_{ij})$, to those
induced by $(\psi_{ij},\Phi_{ij})$ and
$(\psi_{ij}^{\prime},\Phi_{ij}^{\prime})$ (5.27). Thus the corresponding maps
of complexes differ only by a homotopy $h=(h_{1},h_{2})$,

(8.10)
$\textstyle{T_{V_{i}}|_{M_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds_{i}}$$\textstyle{E_{i}^{*}|_{M_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}\\!}$$\scriptstyle{ds_{i}}$$\textstyle{\Omega_{V_{i}}|_{M_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}\\!}$$\textstyle{T_{V_{j}}|_{M_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds_{j}}$$\textstyle{E_{j}^{*}|_{M_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds_{j}}$$\textstyle{\Omega_{V_{j}}|_{M_{ij}}.\\!}$

The $h_{2}$ term is a problem for us because we want to take stupid
truncations (removing the left hand $T_{V}$ terms). The induced (truncated)
maps between the resulting 2-term complexes need not be homotopic unless
$h_{2}=0$. So we next show how to arrange this by lifting homotopies from
virtual cotangent bundles on $M$ to the level of cdgas on $V$.

######  Proposition 8.11.

By altering the equivalence $a_{ij}$ of (8.5) by a homotopy if necessary, we
may assume that $h_{2}=0$.

###### Proof.

We begin by solving this problem in a special case, where the equivalence is
the identity map on the single cdga
$\Lambda^{\bullet}E^{*}\otimes\operatorname{Sym}^{\bullet}(T_{V}[2])$ of (8.4)
— with differential $\delta$ made from a section $s\in H^{0}(V,E)$ and $\nabla
s$ as in (8.4) — and the homotopy $(h_{1},h_{2})$ is $(0,h)$ for some
$h\in\operatorname{Hom}(E^{*}|_{M},T_{V}|_{M})$.

Assume $V$ is Stein and set $M=s^{-1}(0)\subset V$. We show how to modify the
identity cdga map by a homotopy to alter the induced map (8.10) on the virtual
cotangent bundle (which is also the identity of course) by the homotopy
$(0,h)$. Lift $h$ to some

$H\ \in\ \operatorname{Hom}(E^{*},T_{V}).$

Using $\delta$ also for the differential in the virtual cotangent bundle
(8.9), we must define an automorphism of
$\Lambda^{\bullet}E^{*}\otimes\operatorname{Sym}^{\bullet}(T_{V}[2])$,
homotopic to the identity, such that the induced automorphism of its virtual
cotangent bundle (8.9) is $\operatorname{id}+\,[\delta,h]$ on $M$. We use the
identity $\operatorname{id}$ on $\mathcal{O}_{V}$,

(8.12) $f_{t}\ :=\ \operatorname{id}+\,t\hskip 0.6pt[\delta,H]\ =\
\operatorname{id}+\,t\hskip 0.6pt\nabla s\circ H\ \colon\,E^{*}\longrightarrow
E^{*}$

on $E^{*}$, and

(8.13) $g_{t}\ :=\ \operatorname{id}+\,t\hskip 0.6pt[\delta,H]\ =\
\operatorname{id}+\,t\hskip 0.6ptH\circ\nabla s\ \colon\,T_{V}\longrightarrow
T_{V}$

on $T_{V}$, all for fixed $t\in[0,1]$. On the free graded algebra
$\Lambda^{\bullet}E^{*}\otimes\operatorname{Sym}^{\bullet}(T_{V}[2])$ on these
generators, these define the graded algebra endomorphisms

$F_{t}\
:=\,\Lambda^{\bullet}f_{t}\otimes\operatorname{Sym}^{\bullet}(g_{t}[2])\,\colon\,\Lambda^{\bullet}E^{*}\otimes\operatorname{Sym}^{\bullet}(T_{V}[2])\longrightarrow\Lambda^{\bullet}E^{*}\otimes\operatorname{Sym}^{\bullet}(T_{V}[2]).$

Since $[\delta,F_{t}]$ vanishes on the generators
$\mathcal{O}_{V},\,E^{*},\,T_{V}$, the Leibniz rule gives $[\delta,F_{t}]=0$.
Thus the $F_{t}$ are actually cdga endomorphisms.

We claim they in fact fit into a _homotopy_ of cdga automorphisms, i.e. a map
of cdgas

$F_{t}+dt\cdot H_{t}\
\colon\,\Lambda^{\bullet}E^{*}\otimes\operatorname{Sym}^{\bullet}(T_{V}[2])\longrightarrow\Lambda^{\bullet}E^{*}\otimes\operatorname{Sym}^{\bullet}(T_{V}[2])\hskip
0.6pt[t,dt],$

where $t,\,dt$ have degrees 0 and 1 respectively, the differential on the
right hand side is $\delta\otimes 1+1\otimes dt\cdot\frac{d}{dt}(\ \cdot\ )$,
and $\overset{\,{}_{\mbox{\Huge.}}}{F_{t}}=[\delta,H_{t}]$.

To prove the claim we define $H_{t}$ to be $H\colon E^{*}\to T_{V}$ on $E^{*}$
and zero on $\mathcal{O}_{V}$, $T_{V}$. We then extend it inductively to
higher degrees by the formula

(8.14) $H_{t}(ab)\ =\ H_{t}(a)F_{t}(b)+(-1)^{|a|}F_{t}(a)H_{t}(b)$

for homogeneous elements $a,b$ of degrees $|a|,|b|$. To prove
$\overset{\,{}_{\mbox{\Huge.}}}{F_{t}}$ equals $[\delta,H_{t}]$ we check they
agree on homogeneous elements of the cdga by induction on degree. They agree
on the generators $\mathcal{O}_{V},\,E^{*},\,T_{V}$ by (8.12, 8.13). If they
agree on $a$ and $b$ then the Leibniz rule and (8.14) show they both give
$[\delta,H_{t}](a)F_{t}(b)+F_{t}(a)[\delta,H_{t}](b)$ when acting on $ab$.

Thus we get a homotopy from $\operatorname{id}=F_{0}$ to $F_{1}$. By (8.12,
8.13) $F_{1}$ is $\operatorname{id}+\,[\delta,H]$ on $E^{*}$ and $T_{V}$ and
$F_{1}=\operatorname{id}$ on $\mathcal{O}_{V}$. So it induces the map
$\operatorname{id}+\,[\delta,h]$ on the virtual cotangent bundle on $M$, as
required.

We apply this to $V=V_{ij}$ and $M=M_{ij}$ to prove the Proposition. Lift
$h_{2}$ to
$H_{2}\in\operatorname{Hom}\\!\big{(}E_{i}^{*}|_{V_{ij}},T_{V_{j}}|_{V_{ij}}\big{)}$
and set

$H\ :=\ D\big{(}\psi_{ij}^{\prime}\big{)}^{-1}\circ H_{2}\ \in\
\operatorname{Hom}\\!\big{(}E_{i}^{*}|_{V_{ij}},T_{V_{i}}|_{V_{ij}}\big{)}$

so that

(8.15) $(\psi_{ij}^{\prime},\Phi_{ij}^{\prime})+[\delta,H_{2}]\ =\
(\psi_{ij}^{\prime},\Phi_{ij}^{\prime})\circ\Big{(}\\!\operatorname{id}+[\delta,H]\Big{)}.$

The above construction gives a family of homotopic automorphisms $F_{t}$ of
$\Lambda^{\bullet}E_{i}^{*}\otimes\operatorname{Sym}^{\bullet}\\!\big{(}T_{V_{i}}[2]\big{)}\big{|}_{V_{ij}}$
with $F_{0}=\operatorname{id}$ and $F_{1}=\operatorname{id}+\,[\delta,H]$. So
replacing $(\psi_{ij}^{\prime},\Phi_{ij}^{\prime})$ by the homotopic

$(\psi_{ij}^{\prime},\Phi_{ij}^{\prime})\circ
F_{1}\,\stackrel{{\scriptstyle\eqref{homcomp}}}{{=}}\,(\psi_{ij}^{\prime},\Phi_{ij}^{\prime})+[\delta,H_{2}]$

we kill the $h_{2}$ term in the homotopy (8.10) between the induced maps on
virtual cotangent bundles. ∎

Therefore we may assume $a_{ij}$ induces a map of truncated complexes

$\textstyle{E_{i}^{*}|_{M_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds_{i}}$$\textstyle{\Omega_{V_{i}}|_{M_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}\\!}$$\textstyle{E_{j}^{*}|_{M_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds_{j}}$$\textstyle{\Omega_{V_{j}}|_{M_{ij}},\\!}$

just as $(\psi_{ij}^{\prime},\Phi_{ij}^{\prime})$ does, and the two are the
same up to the homotopy $h_{1}$.

Thus the truncation of $a_{ij}$, mapping the truncation
$(\Lambda^{\bullet}E_{i}^{*},s_{i})$ of (8.5) to
$(\Lambda^{\bullet}E_{j}^{*},s_{j})$ — or equivalently mapping between local
holomorphic Kuranishi charts $(V_{i},E_{i},s_{i})$ and $(V_{j},E_{j},s_{j})$
over $V_{ij}$ — is homotopic to $(\psi_{ij}^{\prime},\Phi_{ij}^{\prime})$ in
the sense of Definition 2.10. So the stupid truncation of the description
(8.5) of the derived structure on $M$ defines the complex Kuranishi structure
on $M$ given by the gluing $((\psi_{ij}^{\prime},\Phi_{ij}^{\prime}))$. By
(5.27) this is the Kuranishi structure given by our global complex Kuranishi
chart $(V,E,s)$ (7.10).

$(V,E,s)$ is therefore the output of step (1) of the Borisov-Joyce
construction described in Section 8.1. For step (2) we choose an extension
$E=E^{\mathbb{R}}\oplus iE^{\mathbb{R}}$ over $V$ of the splitting
$E_{1}=E_{1}^{\mathbb{R}}\oplus iE_{1}^{\mathbb{R}}$ on $M$. The latter
satisfies condition $(*)$ of [BJ, Section 3.3] by [BJ, Example 3.8], so the
former does too (at least after shrinking $V$) because $(*)$ is an open
condition [BJ, Theorem 3.7(a)]. Since the restrictions
$E_{i}^{\mathbb{R}}\subset E_{i}$ of $E^{\mathbb{R}}\subset E$ to $V_{i}$ glue
via our $\mathcal{A}_{1}$ transition functions $\Phi_{ij}$ over $V_{ij}$, they
satisfy the compatibility of [BJ, Section 3.4] over $V_{ij}$. So for step (3)
we let $s^{+}$ be the projection of $s$ under $E\to\to E^{\mathbb{R}}$ to get
$(V,E^{\mathbb{R}},s^{+})$ as a global $\mu$-Kuranishi chart for the Borisov-
Joyce $\mu$-Kuranishi structure on $M$.

##  9\. Odd virtual dimension

We are now ready to prove that $[M]^{\operatorname{vir}}_{BJ}$ is torsion when
$\operatorname{vd}$ is odd. First we do some preliminary work with a general
$SO(2n+1,\mathbb{C})$ bundle $(E,q,o)$ on any projective scheme $Y$ with an
isotropic section $s$.282828This will be applied later to the isotropic cone
$C$ (7.8) and the pullback of the special orthogonal bundle $E_{1}$ with its
tautological section $\tau_{E_{1}}|_{C}$. Here $o\in\Gamma(\Lambda^{2n+1}E)$
is an _orientation_ on $(E,q)$ in the sense of [OT, Definition 2.1].

### Maximal isotropics

Suppose first that _$E$ admits a maximal isotropic subbundle_ $\Lambda\subset
E$. It has rank $n$ and is contained in, but is not equal to, its rank $n+1$
annihilator $\Lambda^{\perp}\subset E$ under $q$. We get exact sequences

(9.1) $0\longrightarrow\Lambda\longrightarrow\Lambda^{\perp}\longrightarrow
L\longrightarrow 0\quad\text{and}\quad
0\longrightarrow\Lambda^{\perp}\longrightarrow
E\longrightarrow\Lambda^{*}\longrightarrow 0$

and $L=\Lambda^{\perp}/\Lambda$ inherits a natural quadratic form $q_{L}$ from
$q$. (For even rank $E$ these exact sequences will become the more familiar
(10.3).)

### Orientation

In a local chart in which (9.1) is split, let $\\{e_{i}\\}$ be a local basis
of sections of $\Lambda$ with dual basis $\\{f_{i}\\}$ of $\Lambda^{*}$.

The orientation $o$ on $E$, together with the canonical orientation
$(-i)^{n}e_{1}\wedge f_{1}\wedge\dots\wedge e_{n}\wedge f_{n}$ of [OT,
Equation 18] on $\Lambda\oplus\Lambda^{*}$, defines a canonical orientation
$o_{L}$ on $L$. This makes it an $SO(1,\mathbb{C})=\\{1\\}$ bundle, and
therefore trivial. In fact it is trivialised by the section $o_{L}$ with
$q_{L}(o_{L},o_{L})=1$ whose sign is fixed by

$o\ =\ (-i)^{n}e_{1}\wedge f_{1}\wedge\dots\wedge e_{n}\wedge f_{n}\wedge
o_{L}.$

### Maximal positive definite real subbundles

As in [OT, Section 2.2], $(E,q)$ admits a maximal positive definite real
subbundle $E^{\mathbb{R}}\subset E$, unique up to homotopy. That is,
$E^{\mathbb{R}}$ has real rank $2n+1$ and
$q^{\mathbb{R}}:=q|_{E^{\mathbb{R}}}$ is positive definite (i.e. a real
metric). Then $iE^{\mathbb{R}}\subset E$ is complementary to
$E^{\mathbb{R}}\subset E$, defining a splitting

(9.2) $E\ \cong\ E^{\mathbb{R}}\oplus iE^{\mathbb{R}}\ =\
E^{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}$

with respect to which $q=q^{\mathbb{R}}\otimes\mathbb{C}$. Furthermore (9.2)
defines a complex conjugation operation on $E$.

### Hermitian metric and splittings

Thus $E$ inherits the hermitian metric

$h(\ \cdot\ ,\ \cdot\ )\ :=\ q\big{(}\ \cdot\ ,\overline{\ \cdot\ }\big{)}.$

Then the complex conjugate $\overline{\Lambda}\subset E$ of the maximal
isotropic $\Lambda\subset E$ is also maximal isotropic (but not holomorphic in
general). Via $h$ it gives $\Lambda^{*}\subset E$, which gives a non-
holomorphic splitting of the second exact sequence of (9.1). Finally then,
taking the orthogonal with respect to either $q$ or $h$ (they give the same
result) gives $(\Lambda\oplus\overline{\Lambda})^{\perp}\cong L$ and a
$\mathbb{C}$\- (but not $\mathcal{O}$-) linear splitting of all of (9.1),

(9.3) $E\ \cong\ \Lambda\oplus\Lambda^{*}\oplus L\ =:\ E_{\Lambda}\oplus L.$

With respect to this decomposition, $q$ is the direct sum of $q_{L}$ and the
canonical pairing between $\Lambda$ and $\Lambda^{*}$. Moreover
$E^{\mathbb{R}}$ splits as

$E^{\mathbb{R}}\ \cong\ E_{\Lambda}^{\mathbb{R}}\oplus L^{\mathbb{R}},$

where $L^{\mathbb{R}}=\mathbb{R}.o_{L}$ is the maximal real subbundle of $L$
and $E_{\Lambda}^{\mathbb{R}}$ is the real part of
$E_{\Lambda}=\Lambda\oplus\Lambda^{*}$ — the fixed locus of complex
conjugation.

### Vanishing intersection

Trivialising $(L,q_{L})$ by $o_{L}$, we may write any section of $E$ with
respect to the splitting (9.3) as $s=(e,z)$. Then $s$ is isotropic if and only
if

(9.4) $q(e,e)+z^{2}\ =\ 0.$

Since $q$ is negative definite on $iE^{\mathbb{R}}$, the graph of an isotropic
section $s$ can intersect $iE^{\mathbb{R}}\subset E$ only in the zero section.
We claim the same is true of the homotopic non-isotropic sections

(9.5) $s_{t}\ :=\ (e,tz)\ \in\ \Gamma(E_{\Lambda}\oplus L),\qquad t\in[0,1].$

Indeed if $(e,tz)\in iE^{\mathbb{R}}$ then $e\in iE_{\Lambda}^{\mathbb{R}}$
and $z\in iL^{\mathbb{R}}$ so

$\displaystyle q(e,e)$ $\displaystyle\leq$ $\displaystyle 0\,\text{ with
equality only if }\,e\,=\,0\,\text{ and}$ $\displaystyle z^{2}$
$\displaystyle\leq$ $\displaystyle 0\,\text{ with equality only if
}\,z\,=\,0.$

By (9.4) these imply that $e=0=z$. Therefore, letting $U$ be an
$\epsilon$-neighbourhood of the zero locus $Z(s)\subset Y$ of $s$ inside the
total space of $E\to Y$, we have homology classes

$\big{[}\Gamma_{\\!s_{t}}\big{]}\ \in\ H_{2\dim Y}\big{(}\hskip
0.6pt\overline{U},\partial\overline{U}\\!\smallsetminus
iE^{\mathbb{R}}\big{)}$

and they are all the same for $t\in[0,1]$. We intersect with the Thom class

$[iE^{\mathbb{R}}]\ \in\ H^{2n+1}\big{(}\hskip
0.6pt\overline{U},\partial\overline{U}\\!\smallsetminus
iE^{\mathbb{R}}\big{)}$

defined using the real orientation $o^{\mathbb{R}}$ on its normal bundle (the
pullback of $E^{\mathbb{R}}$) induced from the complex orientation $o$ on $E$
as in [OT, Section 2.2]. This gives

$\Gamma_{\\!s}\cdot iE^{\mathbb{R}}\ =\ \Gamma_{\\!s_{0}}\cdot
iE^{\mathbb{R}}\ \in\ H_{2\dim Y-2n-1}\big{(}\hskip 0.6pt\overline{U}\big{)}\
\cong\ H_{2\dim Y-2n-1}(Z(s)).$

We may then isotop $s_{0}=(e,0)$ to $(e,\delta)$ for some nonzero constant
$\delta\in\mathbb{R}$ to ensure it does not intersect $iE^{\mathbb{R}}$ at
all. We conclude that

(9.6) $\Gamma_{\\!s}\cdot iE^{\mathbb{R}}\ =\ 0.$

### General case

If $E$ does not admit a maximal isotropic subbundle $\Lambda$ then — as in the
even rank case — we pull back to Edidin-Graham’s iterated quadric bundle
$\rho\colon\widetilde{Y}\to Y$ [OT, Equation 19] where there is a tautological
such subbundle $\Lambda_{\rho}\subset\rho^{*}E$. Here (9.6) shows that

$\rho^{*}\Big{(}\Gamma_{\\!s}\cdot iE^{\mathbb{R}}\Big{)}\ =\
\Gamma_{\\!\rho^{*}s}\cdot\rho^{*}iE^{\mathbb{R}}\ =\ 0.$

But if we invert 2 then $\rho^{*}$ has a left inverse
$\frac{1}{2^{n}}\rho_{*}\big{(}h\cup(\ \cdot\ )\big{)}$, where $h\in
H^{2\dim\rho}(\widetilde{M})$ is a distinguished cohomology class [EG,
Proposition 5] whose pushdown $\rho_{*}\hskip 0.6pth=2^{n}$. More generally,
working with $\mathbb{Z}$ coefficients we can conclude the following analogue
of the vanishing (9.6),

(9.7) $2^{n}\big{(}\Gamma_{\\!s}\cdot iE^{\mathbb{R}}\big{)}\ =\ 0$

even without the existence of $\Lambda\subset E$.

### Odd dimensional Borisov-Joyce

To apply this to Calabi-Yau 4-folds, we take $Y$ to be the total space of the
isotropic cone $C\subset E_{1}$ (7.8) with its projection $p\colon C\to M$.
Let $E$ be the bundle $p^{*}E_{1}$ with its tautological isotropic section
$s=\tau_{E_{1}}\big{|}_{C}\,$.

Suppose now that $\operatorname{vd}$ (7.4) is odd. In the notation of (7.6),
$\operatorname{vd}=2\operatorname{rank}T-\operatorname{rank}E_{1}$, so we find
$2n+1:=\operatorname{rank}E_{1}$ is odd. By (9.7) then, the intersection of
$\Gamma_{\\!s}$ and $iE_{1}^{\mathbb{R}}$ is zero when we use
$\textstyle{\mathbb{Z}\big{[}\frac{1}{2}\big{]}}$ coefficients. But by Theorem
8.1 this intersection is the Borisov-Joyce virtual cycle.

######  Theorem 9.8.

Suppose $\operatorname{vd}$ (7.4) is odd. Then the image of the Borisov-Joyce
virtual cycle is zero under the map

$[M]^{\operatorname{vir}}_{BJ}\ \in\
H_{\operatorname{vd}}(M,\mathbb{Z})\longrightarrow
H_{\operatorname{vd}}\big{(}M,\textstyle{\mathbb{Z}\big{[}\frac{1}{2}\big{]}}\big{)}.\vspace{-7mm}$

$\ \hfill\square$

Since Euler classes of $SO(2n+1,\mathbb{R})$ bundles are 2-torsion, it seems
natural to conjecture that $2[M]^{\operatorname{vir}}_{BJ}=0$ when
$\operatorname{vd}$ is odd. Using maximal isotropics we have been able to
prove only that $2^{n}[M]^{\operatorname{vir}}_{BJ}=0$, by (9.7).

Pridham [Pr] has produced a refinement of $[M]^{\operatorname{vir}}_{BJ}$
(tensored with $\mathbb{C}[\\![\hbar]\\!]$ coefficients) using deformation
quantisation. He shows it vanishes when $\operatorname{vd}$ is odd and the
moduli space admits a _global algebraic Darboux chart_ — a global chart (0.1)
in which everything is algebraic and $E$ has a nondegenerate quadratic form
$q$ with respect to which the section $s$ is isotropic: $q(s,s)=0$.

##  10\. tocThe algebraic virtual cycletocThe algebraic virtual cycle

Now let us assume that $\operatorname{vd}$ (0.8) is _even_. Equivalently,
$r=\operatorname{rank}E_{1}$ is even; call it $2n$. The algebraic virtual
cycle of [OT, Definition 4.4] is then

(10.1) $[M]^{\operatorname{vir}}\ =\ \sqrt{0_{E_{1}}^{\,!}\\!}\ [C]\ \in\
A_{\frac{1}{2}\\!\operatorname{vd}}\big{(}M,\mathbb{Z}\big{[}\textstyle{\frac{1}{2}}\big{]}\big{)}$

using the square root Gysin map $\surd\hskip 0.6pt0_{E_{1}}^{\hskip 0.6pt!}$
of [OT, Definition 3.3]. Next we unravel the cosection localisation definition
of this class and use a result of Kiem-Li [KL, Appendix A] to relate it to the
real virtual class (7.15) after inverting 2.

######  Theorem 10.2.

Under the cycle map

$\textstyle{A_{\frac{1}{2}\\!\operatorname{vd}}\big{(}M,\mathbb{Z}\big{[}\frac{1}{2}\big{]}\big{)}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
H_{\operatorname{vd}}\big{(}M,\mathbb{Z}\big{[}\frac{1}{2}\big{]}\big{)},}$$\textstyle{\hskip
25.60747pt[M]^{\operatorname{vir}}\hskip
22.76219pt\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
\ C\cdot iE_{1}^{\mathbb{R}}.\ }$

###### Proof.

Just as in [OT, Section 3] we may assume without loss of generality that
$E_{1}$ admits a maximal isotropic subbundle

(10.3) $0\longrightarrow\Lambda\longrightarrow
E_{1}\longrightarrow\Lambda^{*}\longrightarrow 0,$

which is positive in the sense of [OT, Definition 2.2].292929 By [OT,
Proposition 2.3] this means the isomorphism of real bundles given by the
composition $\Lambda\hookrightarrow E_{1}\cong E_{1}^{\mathbb{R}}\oplus
iE_{1}^{\mathbb{R}}\to\to E_{1}^{\mathbb{R}}$ takes the real orientation on
$\Lambda$ — induced by its complex structure — to the real orientation on
$E_{1}^{\mathbb{R}}$ induced by the complex orientation on $E$ as in [OT,
Section 2.2]. If not we pull back to the cover $\rho\colon\widetilde{M}\to M$
of [OT, Equation 19], where there is a tautological such subbundle
$\Lambda_{\rho}\subset\rho^{*}E_{1}$, and prove the Theorem there. Then we
push the result down to $M$ by $2^{1-n}\rho_{*}(h\cup\ \cdot\ )$, where $h\in
A^{n(n-1)}(\widetilde{Y},\mathbb{Z})$ is a canonical class with
$\rho_{*}h=2^{n-1}$ [OT, Equation 20] (so that $2^{1-n}\rho_{*}(h\cup\ \cdot\
)$ a left inverse of $\rho^{*}$); this is where we need to use
$\mathbb{Z}[1/2]$ coefficients.

We first recall how the class (10.1) is defined in [OT, Definition 3.3]. It
uses [OT, Equation 29] as a way to intersect $C$ with $\Lambda$ inside
$E_{1}$. We spell out the construction. Working on the total space $\pi\colon
E_{1}\to M$, we use the tautological section $\tau_{E_{1}}$ of $\pi^{*}E_{1}$,
restricted to $C\subset E_{1}$. The projection of $\tau_{E_{1}}|_{C}$ to
$\pi^{*}\Lambda^{*}$ has zero locus $\Lambda\cap C$. We linearise about this
zero locus, Fulton-MacPherson style, by degenerating

(10.4) $\Big{(}C\ \subset\ E_{1}\Big{)}\ \rightsquigarrow\
\Big{(}C_{\Lambda\cap C/C}\ \subset\ \Lambda\oplus\Lambda^{*}\Big{)}.$

Here both pairs are isotropic cones inside orthogonal bundles, where on the
right hand side we use the standard quadratic form on
$\Lambda\oplus\Lambda^{*}$ given by pairing $\Lambda$ and $\Lambda^{*}$. This
degeneration is the deformation to the normal bundle of $\Lambda$ in the total
space of $E_{1}$ [Fu, Section 5.1],

(10.5)
---
$\textstyle{\mathcal{E}\ :=\
\mathrm{Bl}_{\Lambda\times\\{0\\}}\big{(}E_{1}\times\mathbb{C}\big{)}\\!\smallsetminus\overline{(E_{1}\times\\{0\\})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{E_{1}}$$\textstyle{\mathbb{C}.\\!}$

The general fibre over $0\neq t\in\mathbb{C}$ is $E_{1}$. The special fibre
over $t=0$ is $\Lambda\oplus\Lambda^{*}$ — the total space of the normal
bundle to $\Lambda\subset E_{1}$. Considering the quadratic form on the fibres
of $E_{1}\to M$ to be a section $q$ of
$\operatorname{Sym}^{2}\Omega_{E_{1}/M}$, we get the nondegenerate quadratic
form

$t^{-1}r^{*}(q)\ \in\ \Gamma\big{(}\operatorname{Sym}^{2}\hskip
0.6pt\Omega_{\mathcal{E}/\mathbb{C}}\big{)}$

on the fibres of (10.5). It is a degeneration of the quadratic form $q$ on the
general fibre to the standard quadratic form on the special fibre
$\Lambda\oplus\Lambda^{*}$. Taking the proper transform of $C\times\mathbb{C}$
in (10.5) gives the degeneration (10.4) of $C\subset E_{1}$ to
$C_{(\Lambda\cap C)/C}\subset\Lambda\oplus\Lambda^{*}$, the normal cone of
$(\Lambda\cap C)\subset C$. Since the generic fibres are all isotropic, so is
this central fibre.

From now on we work on the total space $p\colon\Lambda\to M$. We consider
$\Lambda\oplus\Lambda^{*}$ to be the pullback bundle $p^{*}\Lambda^{*}$,
containing $C_{(\Lambda\cap C)/C}$. To intersect the latter with the 0-section
of the former we use the cosection defined by the tautological section of
$p^{*}\Lambda$,

$\tau_{\Lambda}\,\colon\
p^{*}\Lambda^{*}\longrightarrow\mathcal{O}_{\Lambda}.$

The fact that $C\subset E_{1}$ is isotropic becomes, after the degeneration
(10.5), the identity [OT, Lemma 3.1],

(10.6) $\tau_{\Lambda}\big{|}_{C_{(\Lambda\cap C)/C}}\ \equiv\ 0.$

Thus we may apply [KL, Proposition 1.3] to define a class

(10.7) $0_{p^{*}\Lambda^{*}\\!,\,\tau_{\Lambda}}^{\hskip
0.6pt!,\,\mathrm{loc}}\big{[}C_{(\Lambda\cap C)/C}\big{]}\ \in\
A_{\frac{1}{2}\\!\operatorname{vd}}(M)$

cosection localised to the 0-section $M\subset\Lambda$ — i.e. the zero locus
of the section $\tau_{\Lambda}$ of $p^{*}\Lambda$. In [OT] we defined the
algebraic virtual cycle to be

$[M]^{\operatorname{vir}}\ =\ \sqrt{0_{E_{1}}^{\,!}}\,[C]\ :=\
(-1)^{n}\,0_{p^{*}\Lambda^{*}\\!,\,\tau_{\Lambda}}^{\hskip
0.6pt!,\,\mathrm{loc}}\big{[}C_{(\Lambda\cap C)/C}\big{]}.$

Using [KL, Proposition A.1] this can be given the following more topological
description.

Pick a hermitian metric on $\Lambda$, thus inducing one on $\Lambda^{*}$ and
defining an isomorphism $\Lambda^{*}\cong\overline{\Lambda}$ (where
$\overline{\Lambda}$ is $\Lambda$ as a set and an $\mathbb{R}$-bundle, but
with the conjugate complex structure — i.e. $i$ acts on $\overline{\Lambda}$
by multiplication by $-i$ on $\Lambda$). For a smooth function
$\psi\colon[0,\infty)\to(0,\infty)$ consider the section $\sigma_{\psi}$ of
$p^{*}\overline{\Lambda}\cong p^{*}\Lambda^{*}$ over the total space of
$\Lambda$ which takes $v\in\Lambda$ to $\psi(|v|)v\in
p^{*}\overline{\Lambda}$. Its graph is

(10.8) $\Gamma_{\\!\sigma_{\psi}}\ :=\
\Big{\\{}\big{(}v,\psi(|v|)v\big{)}\,:\,v\in\Lambda\Big{\\}}\ \subset\
\Lambda\oplus\overline{\Lambda}.$

Choosing $0<\delta\ll 1$ and $\psi(|v|)=|v|^{-2}$ for $|v|>\delta$, the
section $\sigma_{\psi}$ satisfies the condition required in [KL, Appendix A],

(10.9) $\langle\sigma_{\psi},\tau_{\Lambda}\hskip 0.6pt\rangle\ =\ 1$

on the complement of the neighbourhood $U^{\delta}_{\Lambda}(M)$ (6.3). Since
$\tau_{\Lambda}$ vanishes on $C_{(\Lambda\cap C)/C}$ (10.6) but equals
$\psi(|v|)|v|^{2}$ on $\Gamma_{\\!\sigma_{\psi}}$, the cone and the graph
intersect only in the zero section $M\subset\Lambda\oplus\overline{\Lambda}$.
Therefore, setting $U=U^{\epsilon}_{\\!\Lambda\oplus\overline{\Lambda}}(M)$,
we can define the topological intersection

(10.10) $\Gamma_{\\!\sigma_{\psi}}\cdot C_{(\Lambda\cap C)/C}\ \in\
H_{\operatorname{vd}}(U)\ \cong\ H_{\operatorname{vd}}(M)$

in the usual way as the cap product of the Thom class

(10.11) $\big{[}\Gamma_{\\!\sigma_{\psi}}\big{]}\ \in\ H^{2n}\big{(}\hskip
0.6pt\overline{U},\partial\overline{U}\\!\smallsetminus\Gamma_{\\!\sigma_{\psi}}\big{)}\quad\text{and}\quad\big{[}C_{(\Lambda\cap
C)/C}\big{]}\ \in\ H_{2\operatorname{rank}T}\big{(}\hskip
0.6pt\overline{U},\partial\overline{U}\\!\smallsetminus\Gamma_{\\!\sigma_{\psi}}\big{)}.$

(Here $T$ is the bundle in the normal form (7.6).) By [KL, Proposition A.1]
the intersection (10.10) is the image of (10.7) under the cycle map
$A_{\frac{1}{2}\\!\operatorname{vd}}(M)\to H_{\operatorname{vd}}(M)$ so

$[M]^{\operatorname{vir}}\ =\ (-1)^{n}\hskip
0.6pt\Gamma_{\\!\sigma_{\psi}}\cdot C_{(\Lambda\cap C)/C}.\vspace{1mm}$

Now deform $\psi$ to the constant function $1$. This deforms the right hand
side of (10.9) from 1 to $|v|^{2}$ through $(0,\infty)$. Thus it is always
nonzero, whereas $\tau_{\Lambda}$ vanishes on $C_{(\Lambda\cap C)/C}$ (10.6),
so again $C_{(\Lambda\cap C)/C}$ and $\Gamma_{\\!\sigma_{\psi}}$ do not meet
outside $M\subset U$. Thus the construction (10.11) of the intersection
(10.10) deforms too, so the homology class of $[M]^{\operatorname{vir}}$ is

$(-1)^{n}\hskip 0.6pt\Gamma_{\\!\sigma_{1}}\cdot C_{(\Lambda\cap C)/C}\ \in\
H_{\operatorname{vd}}(U)\ \cong\ H_{\operatorname{vd}}(M).$

But $\Gamma_{\\!\sigma_{1}}\subset\Lambda\oplus\Lambda^{*}$ is the diagonal in
$\Lambda\oplus\overline{\Lambda}$ — a maximal positive definite real
subbundle. As we deform away from the central fibre of (10.5) back to the
general fibre we can deform
$\Gamma_{\\!\sigma_{1}}\subset\Lambda\oplus\Lambda^{*}$ through other positive
definite subbundles to $E_{1}^{\mathbb{R}}\subset E_{1}$ because the space of
maximal positive definite subspaces of a quadratic vector space is
contractible. As we do so $C_{(\Lambda\cap C)/C}$ deforms back to $C$ through
isotropic cones, which thus miss $E_{1}^{\mathbb{R}}$ outside $M$. Therefore
again the construction (10.11) of the intersection (10.10) deforms too, with
the homology class of $[M]^{\operatorname{vir}}$ becoming

(10.12) $(-1)^{n}E_{1}^{\mathbb{R}}\cdot C\ \in\ H_{\operatorname{vd}}(M).$

Here we have to specify the orientation on the normal bundle
$iE_{1}^{\mathbb{R}}$ to $E_{1}^{\mathbb{R}}\subset E_{1}$ induced by our
construction.

To do so we note $\Lambda\oplus\Lambda^{*}$ has a complex orientation induced
by its deformation (10.5) to $E_{1}$. This induces a canonical real
orientation on the maximal positive definite real subbundle
$\Gamma_{\\!\sigma_{1}}\subset\Lambda\oplus\Lambda^{*}$ by [OT, Section 2.2].
On deformation through the graphs $\Gamma_{\\!\sigma_{\psi}}$ (10.8) back to
$\Gamma_{\\!\sigma_{0}}=\Lambda$ this becomes the standard orientation induced
by the complex structure on $\Lambda$ as in Footnote 29. So once we deform to
$E_{1}^{\mathbb{R}}\subset E_{1}$ we have the canonical real orientation on
$E_{1}^{\mathbb{R}}$ and the canonical real orientation on $E_{1}$ induced by
its complex structure. Together these induce an orientation on the normal
bundle $iE_{1}^{\mathbb{R}}$ to $E_{1}^{\mathbb{R}}\subset E_{1}$ which
differs303030This is the calculation that
$(e_{1}\wedge_{\mathbb{R}}\dots\wedge_{\mathbb{R}}e_{2n})\wedge_{\mathbb{R}}(ie_{1}\wedge_{\mathbb{R}}\dots\wedge_{\mathbb{R}}ie_{2n})$
is equal to
$(-1)^{n}(e_{1}\wedge_{\mathbb{R}}ie_{1})\wedge_{\mathbb{R}}(e_{2}\wedge_{\mathbb{R}}ie_{2})\wedge_{\mathbb{R}}\dots$,
where $\\{e_{i}\\}\subset E_{1}^{\mathbb{R}}$ is an oriented basis for $E_{1}$
in the complex sense, or equivalently for $E_{1}^{\mathbb{R}}$ in the real
sense; see [OT, Section 2.2]. by the sign $(-1)^{n}$ from the one coming from
the isomorphism $\times i\colon E_{1}^{\mathbb{R}}\to iE_{1}^{\mathbb{R}}$.

So applying $\times i$ to $E_{1}$, which preserves $C\subset E_{1}$ and their
real orientations induced by their complex structures, takes (10.12) to

$iE_{1}^{\mathbb{R}}\cdot C\ \in\ H_{\operatorname{vd}}(M).\qed$

Combining Theorem 10.2 with Theorem 8.1 gives

$[M]^{\operatorname{vir}}\ =\ [M]^{\operatorname{vir}}_{BJ}\ \in\
H_{\operatorname{vd}}\big{(}M,\textstyle{\mathbb{Z}\big{[}\frac{1}{2}\big{]}}\big{)},$

thus proving Theorem 4 when $\operatorname{vd}$ is even.

##  Appendix A Some extension results

###  A.1. Stein extension results

We thank Richard Lärkäng and Mohan Ramachandran for expert assistance with the
following examples of Oka’s principle. Recall that by a Stein space we mean a
(not necessarily reduced) complex analytic space which is isomorphic to a
closed analytic subspace of some $\mathbb{C}^{N}$.

######  Proposition A.1.

Given holomorphic maps of analytic schemes

---
$\textstyle{\
X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\textstyle{Y}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$

with $Y$ smooth and $X_{0},\,Y$ Stein, we may shrink $X$ to a Stein
neighbourhood of $X_{0}$ such that there is a a holomorphic map $f\colon X\to
Y$ with $f|_{X_{0}}=f_{0}$.

###### Proof.

Composing $f_{0}$ with an embedding $Y\hookrightarrow\mathbb{C}^{N}$ we get
$N$ holomorphic functions on $X_{0}$ which we can extend to a neighbourhood of
$X_{0}\subset X$. So by shrinking $X$ we may assume we have lifted $f_{0}$ to
$f\colon X\to\mathbb{C}^{N}$.

Since $Y$ is smooth and Stein it has a holomorphic tubular neighbourhood
$Y\,\raisebox{-2.0pt}{$\stackrel{{\scriptstyle\longleftarrow}}{{\hookrightarrow}}$}\,\mathring{\mathbb{C}}^{N}\subset\mathbb{C}^{N}$
by [Siu, Corollary 1]. Shrinking $X$ we may assume $f$ maps it into
$\mathring{\mathbb{C}}^{N}$, then composing with $\mathring{\mathbb{C}}^{N}\to
Y$ gives a map $X\to Y$ extending $f_{0}$. Finally $X_{0}\subset X$ admits a
Stein neighbourhood by the main theorem of [Siu], so we use this to replace
$X$. ∎

######  Proposition A.2.

Fix a $E_{0}$ a holomorphic bundle over $X_{0}\subset Y$, with $X_{0}$ Stein
and $Y$ smooth. Then there is a holomorphic vector bundle $E$ over a Stein
neighbourhood $X\supset X_{0}$ such that $E|_{X_{0}}\cong E_{0}$.

###### Proof.

By [Siu] a Stein neighbourhood $X\supset X_{0}$ exists, and may be embedded
$X\hookrightarrow\mathbb{C}^{N}$. Then there exists a Stein neighbourhood
$\mathring{\mathbb{C}}^{N}$ of $X_{0}$ such that
$X_{0}\subset\mathring{\mathbb{C}}^{N}$ is a deformation retract. Pulling
$E_{0}$ back by the retraction gives a topological bundle $E$ on
$\mathring{\mathbb{C}}^{N}$, which by Grauert’s Oka principle [Fo, Theorem
8.2.1(ii)] admits a holomorphic structure which agrees on
$X_{0}^{\mathrm{red}}$ with $E_{0}|_{X_{0}^{\mathrm{red}}}$.

So we have a bundle $E|_{X}$ over $X$ such that $E|_{X_{0}}$ and $E_{0}$ are
isomorphic on restriction to $X_{0}^{\mathrm{red}}$. What remains is to deduce
that they are isomorphic on $X_{0}$. Filter $X_{0}$ by

$X_{0}^{\mathrm{red}}\ =\ X_{0}^{1}\ \subset\ \dots\ \subset\ X_{0}^{n}\ =\
X_{0}$

so that $X_{0}^{i}\subset X_{0}^{i+1}$ has square zero ideal sheaf $I_{i}$.
Extensions of the bundle $E|_{X_{0}^{i}}$ from $X_{0}^{i}$ to $X_{0}^{i+1}$
form a torsor over $H^{1}\big{(}\mathcal{E}nd\,E|_{X_{0}^{i}}\otimes
I_{i}\big{)}$. Since $X_{0}^{i}$ is Stein this vanishes, so the extension
$E|_{X_{0}^{i+1}}$ is unique. By induction the extension $E|_{X_{0}^{n}}$ is
unique and so equals $E_{0}$. ∎

###  A.2. Extending $\mathcal{A}_{1}$-line bundles

We first prove Theorem 1.24 when $F$ has rank 1. This means its transition
functions commute, so we will be able to use a standard partition of unity
argument on the $C^{\infty}_{V}$-module $I_{1}$.

We cover $V=\bigcup_{i}V_{i}$ by sufficiently small open sets $V_{i}$ that (i)
the $F|_{U_{i}:=V_{i}\cap U}$ are all trivial and (ii) the $V_{i}$ admit Stein
structures compatible with the $\mathcal{A}_{2}$ structure on $V\\!$. That is,
the $C^{\infty}$ structure on $V_{i}$ is that restricted from $V$, and the
two holomorphic structures on $2U\cap V_{i}$ — induced by the Stein structure
on $V_{i}$ and the $\mathcal{A}_{2}$ structure on $V$ — agree.

Thus $F$ is defined by holomorphic transition functions $\phi_{ij}$ over
$U_{ij}$ satisfying $\phi_{jk}\phi_{ik}^{-1}\phi_{ij}\equiv 1$ on $U_{ijk}$.
Shrinking $V_{i}$ if necessary we may lift these to $\mathcal{A}_{1}$
functions $\Phi_{ij}\colon V_{ij}\to\mathbb{C}^{*}$ such that on $V_{ijk}$ the
cocyle

$\delta_{ijk}\ :=\
\Phi_{jk}\Phi_{ik}^{-1}\Phi_{ij}\quad\text{satisfies}\quad\big{|}\delta_{ijk}-1\big{|}\
<\ 1.$

In particular we may use the branch of log on
$\mathbb{C}\,\\!\smallsetminus(-\infty,0]$ which satisfies $\log 1=0$ to
define new smooth transition functions

(A.3) $\Psi_{ij}\ :=\
\exp\left(-\sum\nolimits_{k}h_{k}\log\delta_{kij}\right)\cdot\Phi_{ij}\ \text{
on }V_{ij},$

where the $\\{h_{k}\\}$ form a smooth partition of unity for the cover
$V=\bigcup_{k}V_{k}$. Since $\delta_{ijk}-1\in I_{1}$ and
$\log(1+z)\in(z)\cdot\mathcal{O}$ we see that $\log\delta_{ijk}\in I_{1}$.
Similarly $e^{z}-1\in(z)\cdot\mathcal{O}$ then shows that
$\Psi_{ij}-\Phi_{ij}\in I_{1}$. Thus

$\Psi_{ij}\ \in\ \mathcal{A}_{1}\quad\text{and}\quad\Psi_{ij}\big{|}_{U_{ij}}\
=\ \phi_{ij}.$

Of course the $\Psi_{ij}$ were constructed so that the usual cocycle
argument313131Taking logs in (A.3) turns the unfamiliar multiplicative
argument into the more familiar additive one that a cocyle valued in a fine
sheaf is a coboundary. — using the identity $\delta_{jk\ell}\hskip
0.6pt\delta_{ik\ell}^{-1}\delta_{ij\ell}\hskip 0.6pt\delta_{ijk}^{-1}=1$ —
gives $\Psi_{jk}\Psi_{ik}^{-1}\Psi_{ij}=1$ on $V_{ijk}$. Thus the $\Psi_{ij}$
define $\mathcal{A}_{1}$-transition functions for a line bundle on $V$ which
restricts to $F$ on $U\\!$.

###  A.3. Extending $\mathcal{A}_{1}$-bundles

To prove Theorem 1.24 for a rank $r$ bundle $F$ we may now replace it by its
twist by any holomorphic line bundle on $U$. This extends to an
$\mathcal{A}_{1}$-bundle on $V$ if and only if $F$ does, because we have
already shown the line bundle extends.

So we may assume that $F$ is generated by its global sections, and hence is
the pullback of the universal bundle by a holomorphic map $f\colon
U\to\operatorname{Gr}$ to a Grassmannian $\operatorname{Gr}\hskip
0.6pt(\mathbb{C}^{N},r)$. We will extend $f$ to an
$\mathcal{A}_{1}$-map323232That is $F^{*}\colon
C^{\infty}_{\operatorname{Gr}}\to C^{\infty}_{V}$ maps
$\mathcal{O}_{\operatorname{Gr}}$ in the former to $\mathcal{A}_{1}$ in the
latter. $F\colon V\to\operatorname{Gr}$ in several steps. This will prove
Theorem 1.24, with the holomorphic tranistion functions of the universal
bundle on Gr pulling back to $\mathcal{A}_{1}$-transition functions on $V\\!$.

As before we pick a Stein cover $V=\bigcup_{i}V_{i}$ with holomorphic
extensions $F_{i}\colon V_{i}\to\operatorname{Gr}$ of $f|_{U_{i}}\colon
U_{i}\to\operatorname{Gr}$.

To glue these via a partition of unity argument we need a linear structure on
open sets in Gr. Fix a point $0\to S\to\mathbb{C}^{N}\to Q\to 0$ which we call
$S\in\operatorname{Gr}$ for short. The standard hermitian metric on
$\mathbb{C}^{N}$ defines the orthogonal splitting $\mathbb{C}^{N}\cong S\oplus
Q$. We have the (linear!) open set

$\displaystyle e_{S}\,\colon\,T_{S}\operatorname{Gr}\ \cong\
\operatorname{Hom}(S,Q)$
$\displaystyle\,\lhook\joinrel\relbar\joinrel\rightarrow\,$
$\displaystyle\operatorname{Gr},$ $\displaystyle\phi\quad$
$\displaystyle\longmapsto$
$\displaystyle\operatorname{im}(\operatorname{id}\oplus\hskip 0.6pt\phi),$

where $\operatorname{id}\oplus\hskip 0.6pt\phi\colon S\hookrightarrow S\oplus
Q\cong\mathbb{C}^{N}$. Thus the origin $\phi=0$ maps to
$S\in\operatorname{Gr}$.

This puts a linear structure on the Zariski open set of points
$[S^{\prime}\subset\mathbb{C}^{N}]\in\operatorname{Gr}$ such that
$S^{\prime}\cap S^{\perp}=\\{0\\}$. Allowing the point
$[S\hookrightarrow\mathbb{C}^{N}]$ to vary we get an open neighbourhood of the
diagonal $\Delta\subset\operatorname{Gr}\times\operatorname{Gr}$,

$(\operatorname{Gr}\times\operatorname{Gr})^{\circ}\ :=\
\big{\\{}(S,T)\in\operatorname{Gr}\times\operatorname{Gr}\ \colon\,T\cap
S^{\perp}=\\{0\\}\big{\\}},$

and a $C^{\infty}$ isomorphism

(A.4) $\displaystyle e\,\colon\,T\operatorname{Gr}$
$\displaystyle\xrightarrow{\ \sim\ }$
$\displaystyle(\operatorname{Gr}\times\operatorname{Gr})^{\circ},$
$\displaystyle(S,\phi)$ $\displaystyle\longmapsto$
$\displaystyle\big{(}S,\,\operatorname{im}(\operatorname{id}\oplus\hskip
0.6pt\phi)\big{)}.$

It is holomorphic on each tangent fibre and maps the 0-section to $\Delta$.

Our first (very rough) attempt to glue the $F_{i}$ together is a map $s\colon
V\to\operatorname{Gr}$ which is sufficiently close to $F_{i}$ over $V_{i}$
that each $(s,F_{i})$ maps $V_{i}$ to a neighbourhood of the diagonal
$\Delta$.

######  Lemma A.5.

After possibly shrinking $V$ there exists a $C^{\infty}$ map $s\colon
V\to\operatorname{Gr}$ such that $\big{(}s|_{V_{i}},F_{i}\big{)}\colon
V_{i}\to\operatorname{Gr}\times\operatorname{Gr}$ factors through
$(\operatorname{Gr}\times\operatorname{Gr})^{\circ}$ for all $i$.

###### Proof.

We use the Fubini-Study metric of diameter 1 on Gr and fix $0<\epsilon\ll 1$
so that the $3\epsilon$-neighbourhood $U^{3\epsilon}(\Delta)$ of $\Delta$ is
contained in $(\operatorname{Gr}\times\operatorname{Gr})^{\circ}$.

Consider the $F_{i}$ as (partially defined) sections of the trivial bundle
$V\times\operatorname{Gr}\to V$. Any two of them $F_{i},\,F_{j}$ are
simultaneously defined on $V_{ij}\subset V$ and equal on $U_{ij}\subset
V_{ij}$. Therefore, shrinking $V$ if necessary, we may assume their graphs are
within $\epsilon$ of each other on $V_{ij}$. Thus

$\bigcup\nolimits_{i}\bigcup\nolimits_{x\in V_{i}}\ \\{x\\}\times B_{\hskip
0.6pt3\epsilon}\big{(}F_{i}(x)\big{)}\ \subset\ V\times\operatorname{Gr}$

is a bundle over $V$ with contractible fibres — nonempty unions of balls each
of which contains the centre of the other balls — and local sections
connecting them. It therefore admits a global section $s\colon
V\to\operatorname{Gr}$. ∎

Think of (A.4) as an isomorphism of bundles over the first factor
$\operatorname{Gr}\ni S$. Pulling back by $s$ gives the isomorphism of bundles

$e_{s}\ \colon\,s^{*}(T\operatorname{Gr})\ \xrightarrow{\ \sim\ }\
s^{*}(\operatorname{Gr}\times\operatorname{Gr})^{\circ}$

over $V$. The $F_{i}$ define sections of the latter over $V_{i}$, so we can
now use the linear structure on the fibres of the former — and a partition of
unity $\\{h_{i}\\}$ subordinate to the cover $V=\bigcup_{i}V_{i}$ — to glue
them. This gives the section

(A.6) $F\ :=\ e_{s}\left(\hskip
0.6pt\sum\nolimits_{i}h_{i}\,e_{s}^{-1}(F_{i})\right)\ \text{ of }\
s^{*}(\operatorname{Gr}\times\operatorname{Gr})^{\circ}.$

Here we ignore the $i$th term wherever $h_{i}=0$; since $h_{i}\neq 0$ only on
$V_{i}$ the sum is well defined in $T\operatorname{Gr}$ by Lemma A.5.

The composition of $s^{*}(\operatorname{Gr}\times\operatorname{Gr})^{0}\subset
s^{*}(\operatorname{Gr}\times\operatorname{Gr})\cong V\times\operatorname{Gr}$
with projection to the second factor defines our $C^{\infty}$ map $F\colon
V\to\operatorname{Gr}$.

######  Proposition A.7.

$F\colon V\to\operatorname{Gr}$ is an $\mathcal{A}_{1}$-map with $F|_{U}=f$.

###### Proof.

Using $e$ (A.4) to identify the two bundles
$(\operatorname{Gr}\times\operatorname{Gr})^{\circ}$ and $T\operatorname{Gr}$
over $\operatorname{Gr}$, any map $(a,G)\colon
X\to(\operatorname{Gr}\times\operatorname{Gr})^{\circ}$ induces a map
$e^{-1}\circ(a,G)\colon X\to T\operatorname{Gr}$ which we denote by
$(a,\widetilde{G})$. (So $\widetilde{G}$ is a section of
$a^{*}(T\operatorname{Gr})$.) In this notation, (A.6) becomes

(A.8) $\widetilde{F}\ =\ \sum\nolimits_{i}h_{i}\hskip
0.6pt\widetilde{F}_{i\hskip 0.6pt}.$

Fix any (local) holomorphic function $g$ on $\operatorname{Gr}$. We are
required to show that $F^{*}g\in\mathcal{A}_{1}$. Pulling back $g$ from the
second factor gives $1\otimes g$ on $\operatorname{Gr}\times\operatorname{Gr}$
and so $\widetilde{g}:=e^{*}(1\otimes g)$ on $T\operatorname{Gr}$. These
satisfy

$\widetilde{F}^{*}\hskip 0.6pt\widetilde{g}\ =\
\big{(}e^{-1}\circ(s,F)\big{)}^{*}e^{*}(1\otimes g)\ =\ (s,F)^{*}(1\otimes g)\
=\ F^{*}g.$

So we just need to show that $\widetilde{F}^{*}\hskip
0.6pt\widetilde{g}\in\mathcal{A}_{1}$ locally about a point $x\in V$.
Discarding any $V_{i}$ whose closure does not contain $x$ we may replace $V$
by an open subset of $\bigcap_{i}V_{i}$. Now $F_{1}$ is holomorphic so
$\widetilde{F}_{1}^{*}\hskip 0.6pt\widetilde{g}=F_{1}^{*}g$ is too. Thus it is
sufficient to prove that $F^{*}g-F_{1}^{*}g=\widetilde{F}^{*}\hskip
0.6pt\widetilde{g}-\widetilde{F}_{1}^{*}\hskip 0.6pt\widetilde{g}\in I_{1}$.
(This will also show that $F|_{U}\equiv F_{1}|_{U}\equiv f$, proving the
second claim of the Proposition.) Applying the Hadamard Lemma fibrewise on
$s^{*}(T\operatorname{Gr})\to V$ gives

$\widetilde{F}^{*}\hskip 0.6pt\widetilde{g}-\widetilde{F}_{1}^{*}\hskip
0.6pt\widetilde{g}\ =\
\big{\langle}G,\,\widetilde{F}-\widetilde{F}_{1}\big{\rangle}+\Big{\langle}\,\overline{\\!H},\,\overline{\widetilde{F}-\widetilde{F}_{1}}\,\Big{\rangle}$

for some $T^{*}\operatorname{Gr}$-valued functions $G,H$ made from the
holomorphic (respectively antiholomorphic) derivative of $\widetilde{g}$ down
the fibres of $T\operatorname{Gr}$. But $\widetilde{g}=e^{*}(1\otimes g)$ is
holomorphic on the fibres of $T\operatorname{Gr}$ because $e$ is, so $H\equiv
0$. Thus (A.8) gives

(A.9) $\widetilde{F}^{*}\hskip 0.6pt\widetilde{g}-\widetilde{F}_{1}^{*}\hskip
0.6pt\widetilde{g}\ =\
\Big{\langle}G,\,\sum\nolimits_{i}h_{i}\big{(}\widetilde{F}_{i\hskip
0.6pt}-\widetilde{F}_{1}\big{)}\Big{\rangle}\ =\
\sum\nolimits_{ij}h_{i}\,G_{j}\big{(}\widetilde{F}_{i\hskip
0.6pt}-\widetilde{F}_{1}\big{)}_{j}\hskip 0.6pt.$

For the last equality we have picked a holomorphic trivialisation of
$T\operatorname{Gr}$ over $\operatorname{im}(s)$ (after shrinking $V$ again)
and written $G,\,\widetilde{F}_{i\hskip 0.6pt}-\widetilde{F}_{1}$ in terms of
components. By (A.9) we are left with showing $\big{(}\widetilde{F}_{i\hskip
0.6pt}-\widetilde{F}_{1}\big{)}_{j}\in I_{1}$ for all $i,\hskip 0.6ptj$.

For this we also pick a local _holomorphic_ 333333So not (A.4), which is only
holomorphic on the fibres. trivialisation of the bundle
$(\operatorname{Gr}\times\operatorname{Gr})^{\circ}\to\operatorname{Gr}$ in a
neighbourhood of $\Delta|_{\operatorname{im}(s)}$. Thus we can also express
the $F_{i}$ in terms of components $(F_{i})_{k}$. Then by the Hadamard Lemma
down the fibres of $s^{*}(\operatorname{Gr}\times\operatorname{Gr})^{\circ}\to
V$,

(A.10) $\big{(}\widetilde{F}_{i\hskip 0.6pt}-\widetilde{F}_{1}\big{)}_{j}\ =\
\sum\nolimits_{k}E_{jk}\big{[}(F_{i})_{k}-(F_{1})_{k}\big{]},$

where the $E_{jk}$ are made from the holomorphic derivatives $\partial_{k}$ of
the components of $e^{-1}_{s}$ down the fibres of
$s^{*}(\operatorname{Gr}\times\operatorname{Gr})^{\circ}\to V$. Since
$e^{-1}_{s}$ is fibrewise holomorphic (because $e_{s}$ is) there are no
complex conjugate terms.

Finally then, $F_{i}|_{U}\equiv F_{1}|_{U}$ means $(F_{i})_{k}-(F_{1})_{k}\in
I$, so that (A.10) lies in $I_{1}$ as required. ∎

## References

  * [AMS] M. Abouzaid, M. McLean and I. Smith, _Complex cobordism, Hamiltonian loops and global Kuranishi charts_ , arXiv:2110.14320.
  * [AL] M. Andersson and R. Lärkäng, _The $\overline{\partial}$-equation on a non-reduced analytic space_, Math. Ann. 374 (2019), 553–599. arXiv:1703.01861.
  * [BF] K. Behrend and B. Fantechi, The intrinsic normal cone, Inventiones 128 (1997), 45–88. alg-geom/9601010.
  * [BBBJ] O. Ben-Bassat, C. Brav, V. Bussi and D. Joyce, _A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications_ , Geom. Topol. 19 (2015), 1287–1359. arXiv:1312.0090.
  * [BJ] D. Borisov and D. Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds, Geom. & Top. 21 (2017), 3231–3311. arXiv:1504.00690.
  * [BG] E. Bouaziz and I. Grojnowski, _A d-shifted Darboux theorem_ , arXiv:1309.2197.
  * [BBJ] C. Brav, V. Bussi and D. Joyce, A Darboux theorem for derived schemes with shifted symplectic structure, Jour. AMS 32 (2019), 399–443. arXiv:1305.6302.
  * [CGJ] Y. Cao, J. Gross and D. Joyce, Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds, Adv. Math. 368 (2020), 107134. arXiv:1811.09658.
  * [CL] H.-L. Chang and J. Li, _Semi-perfect obstruction theory and DT invariants of derived objects_ , Comm. Analysis and Geom. 19 (2011), 807–830. arXiv:1105.3261.
  * [De] J.-P. Demailly, _Complex analytic and differential geometry_ , draft book available at https://www-fourier.ujf-grenoble.fr/∼demailly/manuscripts/agbook.pdf.
  * [EG] D. Edidin and W. Graham, Characteristic classes and quadratic bundles, Duke Math. Jour. 78 (1995), 277–299. alg-geom/9412007.
  * [Fo] F. Forstnerič, _Stein manifolds and holomorphic mappings_ , 2nd ed, Springer, 2017.
  * [FOOO] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, _Lagrangian intersection Floer theory: anomaly and obstruction I, II_ , AMS/IP Studies in Advanced Mathematics 46 (2009).
  * [Fu] W. Fulton, Intersection theory, 2nd ed, Springer, 1998.
  * [HT] D. Huybrechts and R. P. Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, Mathematische Annalen 346, 545–569, 2014. arXiv:0805.3527.
  * [Ill] L. Illusie, _Complexe cotangent et déformations I_ , Lecture Notes Math. 239 (1971), Springer.
  * [Jo1] D. Joyce, A new definition of Kuranishi space. arXiv:1409.6908.
  * [Jo2] D. Joyce, _D-manifolds and d-orbifolds: a theory of derived differential geometry_ , draft book available at http://people.maths.ox.ac.uk/∼joyce/dmanifolds.html.
  * [Jo3] D. Joyce, _Algebraic Geometry over $C^{\infty}$-rings_ Memoirs of AMS, Volume 260, 1256 (2019). arXiv:1001.0023.
  * [KL] Y.-H. Kiem and J. Li, Localizing virtual cycles by cosections, Jour. AMS 26 (2013), 1025–1050. arXiv:1007.3085.
  * [KS] Y.-H. Kiem and M. Savvas, _K-Theoretic Generalized Donaldson-Thomas Invariants_ , IMRN 2022 (2022), 2123–2158. arXiv:1912.04966.
  * [LT1] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, JAMS 11 (1998), 119–174. alg-geom/9602007.
  * [LT2] J. Li and G. Tian, Comparison of algebraic and symplectic Gromov–Witten invariants, Asian Jour. Math. 3 (1999), 689–728. alg-geom/9712035.
  * [OT] J. Oh and R. P. Thomas, Counting sheaves on Calabi-Yau 4-folds, I, Duke Math Jour. 172 (2023), 1333–1409. arXiv:2009.05542.
  * [PTVV] T. Pantev, B. Toën, M. Vaquié and G. Vezzosi, Shifted symplectic structures, Publ. Math. IHES 117 (2013), 271–328. arXiv:1111.3209.
  * [Pr] J. Pridham, _Deformation quantisation for $(-2)$-shifted symplectic structures_, arXiv:1809.11028.
  * [Sch] T. Schürg, _Deriving Deligne-Mumford Stacks with Obstruction Theories_ , Geom & Top. 17 (2013), 73–92. arXiv:1005.3945.
  * [Sie] B. Siebert, _Algebraic and symplectic Gromov-Witten invariants coincide_ , Annales de l’Institut Fourier 49 (1999), 1743–1795. math.AG/9804108.
  * [Siu] Y-T. Siu, _Every Stein subvariety admits a Stein neighborhood_ , Inventiones 38 (1976), 89–100.
  * [TVa] B. Toën and M. Vaquie, _Moduli of objects in dg-categories_ , Annales de l’É.N.S. 40 (2007), 387–444. math.AG/0503269.
  * [TVe] B. Toën and G. Vezzosi, _Homotopical algebraic geometry II: Geometric stacks and applications_ , Mem. Amer. Math. Soc. 193 (2008), no. 902. math.AG/0404373.
  * [Vo] B. Volk, _D-orbifolds and d-bordism_ , D. Phil. thesis, Oxford 2014.

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