{\Large \normalfont \scshape \myTitle}
{\tiny \text{\myName}}
We prove a comparison formula for Donaldson-Thomas invariants in the setting of the McKay correspondence. More precisely, we give a relationship between DT-type numbers of a CY3 orbifold with the DT invariants of a given semismall resolution of its coarse moduli space. We then deduce two formulae, identical to Conjectures 1 and 2 of Bryan-Cadman-Young, proving (in the projective case) a twin of the Crepant Resolution Conjecture. Along the way we identify where the standard heart of the orbifold is sent to via the Bridgeland-King-Reid equivalence.
crepresconj.tex
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\newcommand{\myName}{John Calabrese}
\newcommand{\myTitle}{On the Crepant Resolution Conjecture for Donaldson-Thomas Invariants}
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\begin{document}
	\author{{\tiny \text{\myName}}}
	\title{{\Large \normalfont \scshape \myTitle}}
	\address{Mathematical Institute\\ University of Oxford\\ UK}
	\email{john.robert.calabrese@gmail.com}
	\maketitle
	\begin{abstract}
		We prove a comparison formula for Donaldson-Thomas invariants in the setting of the McKay correspondence.
		More precisely, we give a relationship between DT-type numbers of a CY3 orbifold with the DT invariants of a given semismall resolution of its coarse moduli space.
		We then deduce two formulae, identical to Conjectures 1 and 2 of Bryan-Cadman-Young, proving (in the projective case) a twin of the \emph{Crepant Resolution Conjecture}.
		Along the way we identify where the standard heart of the orbifold is sent to via the Bridgeland-King-Reid equivalence.
	\end{abstract}
	\tableofcontents
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intro.tex
\section*{Introduction} % (fold)
\label{sec:introduction}
This paper is concerned with the Crepant Resolution Conjecture for Donaldson-Thomas (DT) invariants as stated by Bryan-Cadman-Young \cite[Conjectures 1 and 2]{crc}.
The goal is to understand the relationship between the DT invariants of a CY3 orbifold $\orb{X}$, satisfying the hard Lefschetz condition, and the DT invariants of a natural crepant resolution $Y \to X$ of its coarse moduli space $X$.
Concretely \eqref{star}, we relate counting invariants of $\orb{X}$ and $Y$.
The proof employs a derived equivalence between $\orb{X}$ and $Y$ (worked out in \cite{chentseng}), which is a ``global'' version of the McKay correspondence of Bridgeland-King-Reid \cite{bkr}.
We prove that the image of the heart $\Coh(\orb{X})$ under this equivalence is Bridgeland's category of perverse coherent sheaves $\Per(Y/X)$ \cite{tomflops}.

Before writing down the formula in symbols, it is profitable to spend a few words on the setup of the conjecture.\footnote{The reader interested in more background on DT theory (and curve-counting in general) could start from \cite{13}.}
Given a smooth and projective Calabi-Yau\footnote{\label{sign}For us \emph{Calabi-Yau} means having trivial canonical bundle $\omega_M \cong \O_M$ and torsion fundamental group $H^1(M,\O_M)=0$.} threefold $M$, we can define the DT invariants of $M$ as weighted Euler characteristics\footnote{There is a minor sign issue in this definition. We expand upon it in Remarks \ref{sign issue crc} and \eqref{sign issue again crc}. A quick inspection will show that the main formulae we prove hold regardless of sign conventions.}
\begin{align*}
	\DT_M(\beta,n) := \chi_\text{top} \left( \Hilb_M(\beta,n), \nu \right) = \sum_{k \in \Z} k \chi_{\topp} \left( \nu^{-1}(k) \right)
\end{align*}
where $\chi_\text{top}$ is the topological Euler characteristic, $\beta \in N_1(M)$ is the homology\footnote{A precise definition will be given in Subsection \ref{reminder}.} class of a curve, $n$ is an integer, $\Hilb_M(\beta,n)$ is the Hilbert (or Quot) scheme parameterising quotients of $\O_M \onto E$ of class $(\ch_0 E, \ch_1 E, \ch_2 E,\ch_3 E) = (0,0,\beta,n)$ and $\nu$ is Behrend's microlocal function \cite{behrend}.%\footnote{This is slightly imprecise as the weight is not given directly by Behrend's function. See Remark}\todo{remark what?} \cite{behrend}.
We formally package these numbers into a generating series.
\begin{align*}
	DT(M) := \sum_{(\beta,n) \in N_1(M) \oplus \Z} DT_M(\beta,n) q^{(\beta,n)}
\end{align*}
Taking Chern characters (and using \cite[Lemma 2.2]{tomcc}) we can we replace $N_1(M) \oplus \Z$ with the numerical Grothendieck group.
To be precise, we let $N(M)$ be the K-group of coherent sheaves on $M$ modulo numerical equivalence and we define $F_1N(M)$ to be the subgroup spanned by sheaves supported in dimension at most one.
It follows that $DT(M)$ can alternatively be indexed by $F_1N(M)$:
\begin{align*}
	DT(M) = \sum_{\alpha \in F_1N(M)} DT_M(\alpha) q^\alpha
\end{align*}
and we will switch between one indexing and the other depending on circumstances. 
There is also a subgroup $F_0N(M)$ spanned by sheaves supported in dimension zero and we can define
\begin{align*}
	DT_0(M) := \sum_{\alpha \in F_0N(M)} DT_M(\alpha) q^\alpha.
\end{align*}

Let now $\orb{X}$ be a projective Calabi-Yau orbifold of dimension three and let $X$ be its coarse moduli space.
By \cite{bkr,chentseng} there is a crepant resolution $Y \to X$ of $X$ given by an appropriate Hilbert scheme of points of $\orb{X}$.
\begin{center}
	\begin{tikzpicture}
		\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em, text height=1.5ex, text depth=0.25ex]
		{
		Y & & \orb{X} \\
		& X & \\
		};
		\path[->,font=\scriptsize]
		(m-1-1) edge node[auto,swap]{$f$} (m-2-2)
		(m-1-3) edge node[auto]{$g$} (m-2-2)
		;
	\end{tikzpicture}
\end{center}
The global McKay correspondence tells us that, moreover, $Y$ and $\orb{X}$ are derived equivalent via Fourier-Mukai transforms
\begin{align*}
	\Phi: D(Y) \rightleftarrows D(\orb{X}):\! \Psi
\end{align*}
inducing isomorphisms between the corresponding (numerical) K-groups.
% \begin{align*}
% 	\phi: N(Y) \rightleftarrows N(\orb{X}): \psi.
% \end{align*}

We also assume that the fibres of $f$ are at most one-dimensional.\footnote{By \cite[Lemma 24]{bg}, a case-by-case analysis shows that this is equivalent to $\orb{X}$ satisfying the hard Lefschetz condition.}
We denote by $F_\text{mr}N(\orb{X})$ the image of $F_1N(Y)$ via $\Phi$ and define\footnote{The subscript $_\text{mr}$ stands for multi-regular, see \cite{crc}.}
\begin{align*}
	DT_\text{mr}(\orb{X}) := \sum_{\alpha \in F_\text{mr}(\orb{X})} DT_\orb{X}(\alpha) q^\alpha
\end{align*}
where the DT number of class $\alpha$ for $\orb{X}$ is defined in the same way, by taking the weighted Euler characteristic of the Hilbert scheme parameterising quotients of $\O_\orb{X}$ of class $\alpha$.
Unfortunately, at this point an ugly technical condition creeps in.
We also need some \emph{partial} DT numbers $\pDT_{\mr}(\orb{X})$ which are given by taking the weighted Euler characteristic of some \emph{open} subspaces of the $\Hilb_\orb{X}(\alpha)$.
We will come back to this subtle point below.
See also Remark \ref{remark}.

The main formula we prove (Corollary \ref{mr=exc}) is
\begin{align}\label{star}\tag{$\bigstar$}
	\pDT_{\mr}(\orb{X}) = \frac{DT_\text{exc}^\vee(Y) DT(Y)}{DT_0(Y)}
\end{align}
where\footnote{The subscript $_\text{exc}$ stands for exceptional locus, notice that in the definition of $DT_\text{exc}$ we only sum over the classes $\beta$ such that $f_*\beta = 0$.}
\begin{align*}
	DT_\text{exc}^\vee(Y) := \sum_{\substack{(\beta,n) \in N_1(Y) \oplus \Z \\ f_* \beta = 0}} DT_Y(-\beta,n)q^{(\beta,n)}
\end{align*}
and where we implicitly identify $F_\text{mr}N(\orb{X})$ with $F_1N(Y)$ via $\Psi$.
The rigorous meaning of this identity will be made precise in Remark \ref{technical}.

To relate the above formula with Conjectures 1 and 2 in \cite{crc} we need two more objects: the subgroup $F_0N(\orb{X})$ of the numerical K-group of $\orb{X}$ spanned by sheaves supported in dimension zero and the subgroup $F_\text{exc}N(Y) \subset F_1N(Y)$ spanned by sheaves whose support is contracted to a point by $f$.
Notice that $F_\text{exc}N(Y)$ is the image of $F_0N(\orb{X})$ via $\Psi$ and $F_\text{exc}N(Y)$ is also the image (via the inverse of the Chern character morphism) of classes $(\beta,n)$ such that $f_* \beta = 0$.
Using this observation one sees (Corollary \ref{zero=exc}) that
\begin{align}\label{conj2}\tag{$\bigstar\bigstar$}
	\pDT_0(\orb{X}) = \frac{DT_\text{exc}(Y)DT_\text{exc}^\vee(Y)}{DT_0(Y)}
\end{align}
where
\begin{align*}
	DT_\text{exc}(Y) = \sum_{\alpha \in F_\text{exc}N(Y)}DT_Y(\alpha)q^\alpha
		= \sum_{\substack{(\beta,n) \in N_1(Y) \oplus \Z \\ f_* \beta = 0}} DT_Y(\beta,n)q^{(\beta,n)}.
\end{align*}
The formula \eqref{conj2} becomes formally identical (with the exception of the superscript $\partial$) to \cite[Conjecture 2]{crc} by using \cite[Theorem 1.1 (b)]{tomcc}.
Plugging \eqref{conj2} in \eqref{star} we obtain
\begin{align*}
	\frac{\pDT_\text{mr}(\orb{X})}{DT_0(\orb{X})} = \frac{DT(Y)}{DT_\text{exc}(Y)}
\end{align*}
which is twin to \cite[Conjecture 1]{crc}.

\subsubsection*{A sketch of the proof} The key result is identifying the image (via $\Psi$) of $\Coh (\orb{X})$  inside $D(Y)$.
It turns out that $\Psi(\Coh(\orb{X}))$ is none other than Bridgeland's heart of perverse coherent sheaves $\Per(Y/X)$.
The relationship between $\Per(Y/X)$ and DT invariants was studied in \cite{cala} (and previously in \cite{todaflops}).
As $\O_Y \in \Per(Y/X)$ one has a \emph{perverse Hilbert scheme} $\PHilb_{Y/X}(\alpha)$ parameterising quotients of $\O_Y$ in $\Per(Y/X)$ of numerical class $\alpha$.
One can then define
\begin{align*}
	DT_{Y/X}(\alpha) := \chi_\text{top}\left( \PHilb_{Y/X}(\alpha) \cap \PHilb_{\leq 1}, \nu \right)
	\quad \text{ and } \quad
	DT(Y/X) := \sum_{\alpha \in F_1(Y)} DT_{Y/X}(\alpha) q^\alpha.
\end{align*}
In \cite{cala} (see also \cite{todaflops}) the following relation between $DT(Y/X)$ and ordinary DT invariants was proved.\footnote{This is slightly imprecise, there is a sign issue which we explain in Remarks \ref{sign issue crc} and \ref{sign issue again crc}.}
\begin{align}\label{flopfo}\tag{$0$}%\tag{$\blacklozenge$}
	DT(Y/X) = \frac{DT_\text{exc}^\vee (Y) DT(Y)}{\DT_0(Y)}
\end{align}
Finally, the Fourier-Mukai transform $\Psi$ not only identifies $\Coh(\orb{X})$ with $\Per(Y/X)$ but also the corresponding Hilbert schemes, so that we have $\Hilb_\orb{X}(\alpha) = \PHilb_{Y/X}(\psi({\alpha}))$.
Again, the technical issue mentioned earlier creeps in.
To define the perverse DT invariants one needs to pass to the open subset of $\PHilb_{Y/X}(\psi(\alpha)) \cap \PHilb_{\leq 1}(Y/X)$, where the latter is the moduli space of quotients of $\O_Y$ in $\Per(Y/X)$ which are supported in dimension at most one.
\emph{A priori}, a quotient $\O_Y \onto P$ might be such that $[P] \in F_1N(Y)$ while $\dim \supp P = \dim \left(\supp H^{-1}(P) \cup \supp H^0(P)\right) > 1$.
We will come back to this in Remark \ref{remark}.
The remedy is to modify the DT invariants on the orbifold side.
For $\psi(\alpha) \in F_1N(Y)$ we define $\Hilb_\orb{X}^\partial(\alpha)$ to be the preimage under $\Psi$ of $\PHilb_{Y/X}(\psi(\alpha)) \cap \PHilb_{\leq 1}(Y/X)$.
The partial DT invariants of $\orb{X}$ are defined by taking the weighted Euler characteristics of these latter moduli spaces.
We then obtain
\begin{align*}
	\pDT_\text{mr}(\orb{X}) = DT(Y/X)
\end{align*}
which implies \eqref{star}.

We mention in passing that the arguments contained in this paper morally work just as well in the quasi-projective setting.
Unfortunately, however, the identity \eqref{flopfo} is currently only available for $Y$ projective.
After the first version of this paper was uploaded on the arXiv, an article by David Steinberg \cite{david} also appeared.
Steinberg has been laying siege to the same conjecture using an interesting \emph{relative stable pairs} approach, which involves a different t-structure.

\subsubsection*{Acknowledgements} The author would like to thank Tom Bridgeland for precious help, Dominic Joyce for useful conversations, Arend Bayer for sharing a preliminary version of an upcoming paper and Jim Bryan and David Steinberg for sharing their ideas.

\subsubsection*{Structure of the paper} The paper is divided into two sections.
The first one is the core, as it contains the proof of the fact that $\Coh (\orb{X})$ is sent to $\Per(Y/X)$ via the derived equivalence.
In the second section we apply this result to Donaldson-Thomas invariants.

\subsubsection*{Conventions} We work over the field of complex numbers $\C$. For a scheme (or stack) $M$, $D(M)$ will denote the \emph{bounded} derived category of coherent $\O_M$-modules.

% section introduction (end)

















macros.tex
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one.tex
\section{The Equivalence between $\Per(Y/X)$ and $\Coh (\orb{X})$} % (fold)
\label{sec:the_equivalence_between_per_y_x_and_coh_x_}
We work in the following setup.
\begin{situ}\label{situ1}
	Let $\orb{X}$ be a smooth, quasi-projective, Deligne-Mumford stack of dimension $n$.
	Assume the canonical bundle $\omega_{\orb{X}}$ to be Zariski-locally trivial and denote by $X$ the coarse moduli space of $\orb{X}$.
\end{situ}
\begin{rmk}
	The bundle $\omega_\orb{X}$ on $\orb{X}$ is \emph{Zariski}-locally trivial if there exists a Zariski open cover $\orb{X}' \to \orb{X}$ (where we allow $\orb{X}'$ to be a \emph{stack}) such that the restriction $\omega_\orb{X}\vert \orb{X}'$ is trivial.
	This is a technical condition which, by working locally on the coarse space $X$, allows us to reduce to the setting of \cite{bkr}.
	In fact, in the case where $\orb{X} = [V/G]$, it amounts to requiring that the canonical bundle of $V$ be $G$-equivariantly locally trivial.
	This condition seems to be missing in \cite{chentseng}.
\end{rmk}

It is beneficial to recall the framework of \cite{chentseng}.
A candidate for a resolution of $X$ (and a replacement for the equivariant Hilbert scheme found in \cite{bkr}) is given by the irreducible component $Y$ of the Hilbert scheme $\Hilb (\orb{X})$ containing the non-stacky points of $\orb{X}$.\footnote{It is probably helpful to remark that for a stack $\orb{X}$ there might be some ambiguity in the term \emph{Hilbert scheme} (see \cite{rydh}). However, we shall always interpret Hilbert schemes as Quot functors, which for Deligne-Mumford stacks were studied by Olsson and Starr \cite{os}.}
The morphism $g\colon \orb{X} \to X$ induces a morphism $\Hilb(\orb{X}) \to \Hilb(X)$ and, by restriction, a morphism $f\colon Y \to X$. We draw a diagram.
\begin{center}
	\begin{tikzpicture}
		\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em, text height=1.5ex, text depth=0.25ex]
		{
		& Y \times \orb{X} &\\
		Y & &\orb{X}\\
		& X &\\
		};
		\path[->,font=\scriptsize]
		(m-2-1) edge node[auto,swap]{$f$} (m-3-2)
		(m-2-3) edge node[auto]{$g$} (m-3-2)
		(m-1-2) edge node[auto,swap]{$\pi_Y$} (m-2-1)
				edge node[auto]{$\pi_\orb{X}$} (m-2-3)
		;
	\end{tikzpicture}
\end{center}
%\begin{rmk}%\label{rema}
	Under the additional assumption that $Y\times_X Y$ is at most of dimension $n+1$ it is proved in \cite{chentseng} that $Y$ is smooth and that $f$ is a crepant resolution.
	Furthermore, the scheme $Y$ represents a moduli functor and its corresponding universal object is a quotient $\O_{Y \times \orb{X}} \onto \O_\orb{Z}$.
	Finally, it is shown that one has a Fourier-Mukai equivalence $D(Y) \simeq D(\orb{X})$ with kernel given by $\O_\orb{Z}$.
	% \begin{align*}
	% 	\Phi\colon D(Y) \to D(\orb{X})
	% \end{align*}
	
	We recall three key results involved in the proof: the Hilbert scheme $\Hilb (\orb{X})$ commutes with \'etale base-change on $X$ \cite[Proposition 2.3]{chentseng}; \'etale-locally on $X$ the space $\orb{X}$ is isomorphic to a quotient stack $[V/G]$, with $V$ smooth and affine and $G$ a finite group (whose coarse space is thus the quotient $V/G$) \cite[Lemma 2.2.3]{vista}; the Hilbert scheme of $[V/G]$ is isomorphic to Nakamura's $G$-equivariant Hilbert scheme $G\text{-}\Hilb (V)$ \cite[Lemma 2.2]{chentseng}.
	Exploiting these facts one may reduce to \cite{bkr}, as checking that the given kernel produces an equivalence may be done locally \cite[Proposition 3.3]{chentseng}.
%\end{rmk}
\begin{rmk}\label{otherperversity}
	As is usual with integral transforms, the kernel $\O_\orb{Z}$ may be interpreted as giving a functor in two different directions.
	The standard Mukai-implies-McKay convention is to take $\O_\orb{Z}$ to define a functor $\twit{\Phi}\colon D(Y) \to D(\orb{X})$ \cite{bkr,chentseng}.
	To deal with a technical issue (caused by \cite{cala}), we will also consider $\hat{\Phi} = \D\Phi\D \colon D(Y) \to D(\orb{X})$, where $\D = \R\underline{\Hom}(-,\O)$ is the duality functor.
	We denote by $\Psi$ the inverse of $\Phi$ and by $\hat{\Psi}$ the inverse of $\hat{\Phi}$.
	When $Y$ and $X$ are projective, the relationship between $\Phi$ and $\hat{\Phi}$ is quite simple, as $\hat{\Psi}$ is given by the Fourier-Mukai transform with kernel $\O_\orb{Z}$ (this is a standard consequence of \cite[Propositions 1.13 and 1.15]{nahm}).
	% \qui $\otwit{\Psi}\colon D(\orb{X}) \to D(Y)$, the Fourier-Mukai transform with kernel $\O_\orb{Z}$ but going in the opposite direction.
	% The relationship with $\twit{\Phi}$ is quite simple, as $\otwit{\Psi} = \D\twit{\Phi}^{-1}\D,$ where $\D = \R \underline{\Hom}(-,\O)$ is the duality functor (this is a standard consequence of \cite[Propositions 1.13 and 1.15]{nahm}).
	% We shall denote by $\otwit{\Phi}$ the inverse of $\otwit{\Psi}$, and by $\twit{\Psi}$ the inverse of $\twit{\Phi}$.
	% Notice that both $\otwit{\Phi}$ and $\twit{\Phi}$ are functors from $Y$ to $\orb{X}$.
\end{rmk}
We now briefly remind the reader of Bridgeland's heart of perverse coherent sheaves \cite{tomflops}.
In some sense, it is a reflection of the ambiguity revolving around the kernel $\O_\orb{Z}$ that we consider both the $-1$ and $0$ perversity.
The category $\perv{p}{\Per(Y/X)}$ of \emph{perverse coherent} of \emph{perversity} $p \in \{-1,0\}$ consists of those complexes $E \in D(Y)$ satisfying
\begin{itemize}
	\item $\R f_* E \in \Coh (X)$,
	\item $\Ext^{-i}_Y (E, C) = 0 = \Ext_Y^{-i}(C,E)$, for all $i > p$ and all $C \in \Coh (Y)$ such that $\R f_* C = 0$.
\end{itemize}

The rest of this section is devoted to the proof of the following statement.
\begin{thmm}\label{hearts}
	Assume to be working in Situation \ref{situ1} and assume in addition $f$ to have relative dimension at most one.
	Then the equivalence $\Phi$ between $D(Y)$ and $D(\orb{X})$ restricts to an equivalence of abelian categories between $\perv{0}{\Per}(Y/X)$ and $\Coh (\orb{X})$, while the equivalence $\otwit{\Phi}$ restricts to an equivalence between $\perv{-1}{\Per(Y/X)}$ and $\Coh(\orb{X})$.
\end{thmm}
\begin{rmk}
	Notice that the condition $\dim Y \times_X Y \leq n+1$ follows automatically from the condition on the fibres of $f$.
\end{rmk}
In particular $\perv{0}{\Per(Y/X)}$ is equivalent to $\perv{-1}{\Per(Y/X)}$.
We also point out that the composition $\otwit{\Phi}\twit{\Phi}^{-1}$ gives a non-trivial autoequivalence of $D(\orb{X})$, which seems related to the \emph{window shifts} of Donovan-Segal \cite{window}.
It might be worthwhile to compute this equivalence in explicit examples.

Let us now begin the proof of the theorem, which will be divided into small steps.
We start by considering $\twit{\Phi}$.
% Let us momentarily drop the $^0$ superscript from $\qerv{0}{\Per(Y/X)} = \Per(Y/X)$.\
\smallskip
\stepz Given an object of the derived category, membership of either of the categories in question can be checked \'etale-locally on $X$ \cite[Proposition 3.1.6]{vdb}.
Thus, by base-changing over \'etale patches of $X$, we can reduce to the case where $X$ is affine and furthermore $\orb{X} = [V/G]$ with $V$ a smooth affine scheme and $G$ finite.
Moreover, the functors $\twit{\Phi}$ and $\twit{\Psi}$ (being Fourier-Mukai) commute with this base-change \cite[Proposition 6.1]{nahm}.

\smallskip
\stepz It suffices to prove $\twit{\Psi}(\Coh (\orb{X})) \subset \qerv{0}{\Per}(Y/X)$ because of the following well-known trick.
\begin{lem}
	Let $\curvy{A}$ and $\curvy{B}$ be two hearts relative to two bounded t-structures in a triangulated category.
	Then $\curvy{A} \subset \curvy{B}$ if and only if $\curvy{B} \subset \curvy{A}.$
\end{lem}
\begin{prf}
	Given an object $E$ let us denote by $H_\curvy{A}^i(E)$ (respectively $H_\curvy{B}^i(E)$) the $i$-th cohomology object relative to $\curvy{A}$ (resp.~$\curvy{B}$).
	Assume $\curvy{A} \subset \curvy{B}.$
	Let $E \in \curvy{B}$.
	As $E$ already lies in $\curvy{B}$ we have $E \simeq H^0_\curvy{B}(E)$ and $H^i_\curvy{B}(E) = 0$ for $i \neq 0.$
	Consider now the cohomology filtration of $E$ relative to $\curvy{A}$.
	As objects of $\curvy{A}$ are also in $\curvy{B}$, this filtration is also a filtration relative to $\curvy{B}$.
	By uniqueness of the cohomology objects we have $H^i_\curvy{A}(E) = H^i_\curvy{B}(E) = 0$ for $i\neq 0.$
	Thus, $E \in \curvy{A}$.
\end{prf}
\smallskip
\stepz To prove the mentioned inclusion we will exhibit two systems of generators (see definition below), one for $\qerv{0}{\Per}(Y/X)$ and one for $\Coh (\orb{X})$, and show that elements of the first system are sent to the second.
\begin{defn}
	Let $\cat{D}$ be a triangulated category and let $\curvy{A}$ be the heart of a bounded t-structure.
	A collection $\curly{P}$ of objects of $\curvy{A}$ is a \emph{system of projective generators} if, for all $A \in \curvy{A} \setminus \{0\}$ and all $P \in \curly{P}$, $\Ext_\cat{D}^\bullet(P,A)$ is concentrated in degree zero and for all $A \in \curvy{A}$ there exists $P_A \in \curly{P}$ such that $\Hom_\cat{D}(P_A,A) \neq 0$.%\footnote{We remind ourselves that in any triangulated category one can define $\Ext^n_\cat{D}(P,A) := \Hom_\cat{D}(P,A[n])$, where $[1]$ is the shift functor.}
\end{defn}
By \cite[Lemma 3.2.4]{vdb}, when $X$ is affine, we have a system of generators $\curly{P}$ for $\qerv{0}{\Per}(Y/X)$ consisting of vector bundles $P$ such that
\begin{itemize}
	\item $\R^1f_*P = 0$,
	\item $P^\vee$ is generated by global sections.
\end{itemize}
For $\Coh (\orb{X})$ we also have a nice system of generators.
\begin{lem}
	The collection $\curly{Q}$ of vector bundles on $\orb{X}$ is a system of generators for $\Coh (\orb{X})$.
\end{lem}
\begin{prf}
	As we are working in the case $\orb{X} = [V/G]$, it is easy to reduce the problem to bundles on $V$.
	In fact, coherent sheaves on $\orb{X}$ are $G$-equivariant coherent sheaves on $V$.
	Given an equivariant vector bundle $P$ and an equivariant sheaf $E$ on $V$ we have that $G\text{-}\Ext^i_V(P,E) = \Ext_V^i(P,E)^G$, where the latter is the $G$-invariant part \cite[Section 4.1]{bkr}.
	As $V$ is affine, these groups vanish for $i>0$.
	
	Fix now an equivariant sheaf $E$, we want to find an equivariant vector bundle $P$ such that $\Hom_V(P,E)^G \neq 0$.
	By \cite[Lemma 4.1]{bkr} $\Hom_V(P,E)$ splits as a direct sum of $\Hom_V(P\otimes \rho, E)^G \otimes \rho$, where $\rho$ ranges among the irreducible representations of $G$.
	The claim thus follows as $P \otimes \rho$ is a vector bundle.
\end{prf}

%\smallskip
\stepz We now conclude the proof by showing that elements of $\curly{P}$ are sent to elements of $\curly{Q}$.
First we remark that we can check whether a complex on $\orb{X} = [V/G]$ is a vector bundle by pulling back to the \'etale atlas $V \to [V/G]$.
Thus, if $P \in \curly{P}$, we are interested in the pullback of $\twit{\Phi}(P)$ to $V$.
This allows us to reduce to the setup of \cite{bkr}, where one has the following diagram.
\begin{center}
	\begin{tikzpicture}
		\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em, text height=1.5ex, text depth=0.25ex]
		{&Z&\\
		&Y\times V&\\
		Y&&V\\
		&X&\\
		};
		\path[->,font=\scriptsize]
		(m-1-2) edge[bend right=20] node[auto,swap]{$p$}(m-3-1)
				edge[bend left=20] node[auto]{$q$}(m-3-3)
		(m-1-2) edge node[auto]{$i$} (m-2-2)
		(m-2-2) edge node[auto,swap]{$\pi_Y$} (m-3-1)
				edge node[auto]{$\pi_V$} (m-3-3)
		(m-3-1) edge node[auto,swap]{$f$}(m-4-2)
		(m-3-3) edge node[auto]{$g$}(m-4-2)
		;
	\end{tikzpicture}
\end{center}
Here $Z$ is the universal $G$-cluster for the action of $G$ on $V$, $q$ and $f$ are proper and birational, $p$ and $g$ are finite and $p$ is also flat.
Moreover, the quotient $\O_{Y \times V} \onto \O_Z$ is precisely the pullback, under the morphism $Y \times V \to Y \times [V/G] = Y \times \orb{X}$, of the universal quotient $\O_{Y \times \orb{X}} \onto \O_\orb{Z}$, which we used to define $\twit{\Phi}$.
It follows that applying $\twit{\Phi}$ followed by pulling back to $V$ is the same as applying $\R q_* p^*$.

We have thus reduced our final step to checking that, given an element $P \in \curly{P}$, the complex $\R q_* p^* P$ is actually a vector bundle.
\begin{lem}
	Let $P \in \Coh(Y)$ satisfy $\R^1 f_* P = 0$.
	Then $\R q_* p^* P \in \Coh(\orb{X})$.
\end{lem}
\begin{prf}
	Notice that $\R q_* p^* P = \R \pi_{V,*} i_* p^* P = \R \pi_{V,*} (\pi_Y^* P \otimes \O_Z)$, where we made the standard identification $\O_Z = i_* \O_Z$.
	We point out that, as a consequence of our assumption on $f$, $\pi_{V,*}$ is of homological dimension at most one (we remind the reader that we work under the reduction done in the Step 1, in particular $X$ is affine).
	
	By tensoring the quotient $\O_{Y \times V} \onto \O_Z$ with $\pi_Y^*P$ we produce a surjection $\pi_Y^*P \onto \pi_Y^* P \otimes \O_Z$.
	Applying $\pi_{V,*}$ yields a surjection $\R^1\pi_{V,*} \pi_Y^* P \onto \R^1 \pi_{V,*} (\pi_Y^* P \otimes \O_Z)$.
	But $\R^1 \pi_{V,*} \pi_Y^* P = H^1(Y,P) \otimes_\C \O_V$ and $H^1(Y,P) = 0$ as $\R^1f_* P = 0$, hence the claim.
\end{prf}
\begin{lem}
	Let $P \in \curly{P}$, then $\R q_* p^* P$ is a vector bundle on $V$.
\end{lem}
\begin{prf}
	We know that the dual of $P$ is generated by global sections, hence there exists a short exact sequence
	\begin{align*}
		K \into \O^{\oplus m}_Y \onto P^\vee.
	\end{align*}
	From the fact that $P$ and $\O_Y$ are vector bundles it follows that $K$ is also a vector bundle.
	We therefore have a dual sequence
	\begin{align*}
		P \into \O^{\oplus m}_Y \onto K^\vee.
	\end{align*}
	It follows from the previous lemma, plus the fact that $q_* \O_Z = \O_V$, that applying $\R q_* p^*$ yields an exact sequence
	\begin{align*}
		q_* p^* P \into \O^{\oplus m}_V \onto q_* p^* K^\vee.
	\end{align*}
	To prove our claim it suffices to check that $\Ext^1_V(q_*p^*P,M) = 0$ for all modules $M$ on $V$.
	By the above short exact sequence this is the same as showing that $\Ext^2_V(q_*p^* K^\vee,M) = 0$ for all modules $M$.
	Using Grothendieck duality %\begin{comment}\cite{rd,lipman}\end{comment}
	for $q$ we have
	\begin{align*}
		\Ext_V^2(q_*p^* K^\vee,M)
		= \Ext_Z^2(p^*K^\vee,q^!M) 
		= H^2(Z,p^*K \otimes q^! M).
		% & = \Ext_V^2(p^*K^\vee, \L q^* M \ltensor \omega_Z) \\
		% & = H^2(Z,p^*K \otimes \L q^* M \ltensor \omega_Z)
	\end{align*}
	The scheme $Z$ admits a finite and flat map to a smooth variety ($f: Z \to Y$) thus it is Cohen-Macaulay.
	Moreover, as $\dim Z - \dim V = 0$ and $q$ is of finite tor-dimension, the complex $q^!M$ is concentrated in non-positive degrees.
	As our assumption on $f$ implies that $H^i(Z,E) = 0$ for all $i > 1$ and all sheaves $E$, the hypercohomology spectral sequence tells us that $H^2(Z,p^*K \otimes q^!M) = 0$.
	Hence we are done.
\end{prf}
The previous lemma concludes the first half of the proof.
As is often the case, the second half is much shorter than the first.
In fact, to prove the statement for $\perv{-1}{\Per(Y/X)}$ and $\Phi$, one need only notice the following:
\begin{itemize}
	\item $\otwit{\Phi} = \D \twit{\Phi} \D$, %(as explained in Remark \ref{otherperversity}),
	\item the dual system $\curly{P}^\vee = \D \curly{P}$ is a system of generators for $\perv{-1}{\Per(Y/X)}$ \cite[3.2.3]{vdb},
	\item the system $\curly{Q}$ is self-dual $\D \curly{Q} = \curly{Q}$.
\end{itemize}
This concludes the proof and we can now move on to comparing the DT invariants of $\orb{X}$ and $Y$.
\begin{rmk}\label{o}
	For the next section, it will be important to know that $\Phi(\O_Y) = \O_\orb{X}$.
	We already know that $\Phi(\O_Y)$ is a vector bundle given by $\R q_*\O_\orb{Z}.$
	By restricting to the smooth locus of $X$ (viz.~to an open where $\Phi$ is the identity) we see that $\R q_* \O_\orb{Z}$ is in fact a line bundle.
	In turn this implies that $\Phi(\O_Y) = \O_\orb{X}$ as the unit $\O_\orb{X} \to \R q_* q^* \O_\orb{X}$ is an isomorphism.
	The same statement obviously holds for $\hat{\Phi} = \D \Phi \D$ as well.
\end{rmk}
% 	
% 	
% 	It can be useful to know that the equivalences described above commute with pushing down to $X$.
% 	For example, to check $g_* \twit{\Phi} = \R f_*$, one may start by observing that $\Hom_Y(\pi_{Y,*}\O_\orb{Z}, \O_y[i]) = $
% 	
% 	\qui
% 	one may start by using the projection formula to reduce the problem to proving\tom{to reducing to prove: yuck.} that $\pi_{Y,*} \O_{\orb{Z}} = \O_Y$.
% 	By computing $\Hom_$
% 	But $\pi_{Y,*}\O_\orb{Z}$ is a line bundle with a nowhere vanishing section induced by $\O_{Y \times \orb{X}} \onto \O_\orb{Z}$.\tom{why???}
% \end{rmk}
% 	\todo[inline]{does line bundle come from mapping to points? LMB remarks that if pushdown of SS is LB then the standard map $O \to p_*O$ is an iso (it's a local question).}
\begin{rmk}\label{usefulcommute}
	It can be useful to know that when $Y$ and $X$ are projective the equivalences described above commute with pushing down to $X$.
	For example, let us check that $g_* \Phi = \R f_*$.
	We have $g_* \Phi = \R f_* \R p_* p^*$.
	If we proved that $\R p_* \O_\orb{Z} = \O_Y$, then by the projection formula we would be done.
	Thankfully, the previous remark together with Remark \ref{otherperversity} already tell us that $\R p_* \O_\orb{Z} = \Psi(\O_\orb{X}) = \O_Y$.
	% 
	% 
	% \qui
	% For the next section, it will be important to know that $\Phi(\O_Y) = \O_\orb{X}$, which is equivalent to $\Psi(\O_\orb{X}) = \O_Y$.
	% One first checks that $\Psi(\O_\orb{X})$ is a vector bundle.
	% This can be done locally on $X$, so we can use the same reduction steps as in the proof above, and moreover it suffices to show that $\Ext_Y^\bullet(\Psi(\O_\orb{X}),\O_y)$ is concentrated in degree zero, for any closed point $y \in Y$.
	% Using the equivalence this is the same as $\Ext^\bullet_{\orb{X}}(\O_\orb{X},\Phi(\O_y))$, which satisfies the required vanishing as $\Phi(\O_y)$ is a sheaf.\tom{tom adds: supp'd in dim zero, but I don't need this as I'm affine}
	% By inspecting $\Psi(\O_\orb{X})$ away from the exceptional locus one sees that it is indeed a line bundle.
	% By the previous remark, we know that $\R f_* \Psi(\O_\orb{X}) = g_* \O_\orb{X} = \O_X$.
	% In particular (by adjunction), the line bundle $\Psi(\O_\orb{X})$ has a non-zero section.
	% As $f$ is crepant, we also deduce (using duality) that $\R f_* \Psi(\O_\orb{X})^\vee = \O_X$ and so $L^\vee$ also admits a non-zero section.
	% Thus $\Psi(\O_\orb{X})$ must be the trivial bundle.
	% The analogous fact holds for $\otwit{\Phi}$.
\end{rmk}

% section the_equivalence_between_per_y_x_and_coh_x_ (end)
















two.tex
\section{The Formula for DT Invariants} % (fold)
\label{sec:the_formula_for_dt_invariants}
We now impose further restrictions on our spaces.
\begin{situ}\label{situ2}
	Recall Situation \ref{situ1} and assume in addition $\orb{X}$ to be projective and of dimension three.
	Assume moreover $\orb{X}$ to be Calabi-Yau, i.e.~$\omega_\orb{X} \cong \O_\orb{X}$ and $H^1(\orb{X},\O_\orb{X})=0$.
	Finally, assume the crepant resolution $f\colon Y \to X$ of the previous section to have relative dimension at most one.
\end{situ}
\begin{rmk}
	We follow the convention where a Deligne-Mumford stack is projective if its coarse moduli space is.
	From the assumptions above it follows that $X$ is of dimension three, projective, Gorenstein with quotient singularities and with trivial canonical bundle.
	In turn it follows that $Y$ is Calabi-Yau of dimension three and that $X$ has rational singularities, and so $\R f_*\O_Y=\O_X$ \cite{kovacs}.
\end{rmk}
As the functor $\twit{\Phi}$ is more natural from the perspective of the McKay correspondence we shall focus on the zero perversity.
\begin{notation}
	We will drop the superscript $^{0}$ from $\perv{0}{\Per(Y/X)} =: \Per(Y/X)$.
\end{notation}

\addtocontents{toc}{\protect\setcounter{tocdepth}{0}}
\subsection{Reminder}\label{reminder}
Let us recall some definitions from the introduction.
We denote by $N(Y)$ the \emph{numerical K-group} of coherent sheaves of $Y$.
We remind ourselves that we can define a bilinear form on $K_0(\Coh (Y))$
\begin{align*}
	\chi(E,F) := \sum_k (-1)^k \dim_\C \Ext^k_Y(E,F)
\end{align*}
and that $N(Y)$ is obtained by quotienting out its radical.
Inside $N(Y)$ we can single out $F_1N(Y)$, which is the subgroup generated by sheaves supported in dimensions at most one.
We also define $F_{\exc}N(Y)$ to be the subgroup of $F_1N(Y)$ spanned by sheaves with derived pushforward to $X$ supported in dimension zero.

To $Y$ one can also attach the numerical Chow groups $N_*(Y)$, which are the groups of cycles modulo numerical equivalence.
We write $N_{\leq 1}(Y) := N_1(Y) \oplus N_0(Y)$ and recall that $N_0(Y) \cong \Z$.
In \cite[Lemma 2.2]{tomcc} it is shown that the Chern character induces an isomorphism $F_1N(Y) \cong N_{\leq 1}(Y) \cong N_1(Y) \oplus \Z$, which allows us to pass from one group to the other.
Using this identification, $F_{\exc}N(Y)$ can be rewritten as
\begin{align*}
	F_{\exc} N(Y) = \left\{ (\beta,n)\in N_1(Y) \oplus \Z \st f_*\beta=0 \right\}
\end{align*}
where $f_*$ here stands for the proper pushforward on cycles (the subscript $\exc$ is short for \emph{exceptional}).

For the orbifold $\orb{X}$ we can also define a numerical K-group $N(\orb{X})$.
Inside it lies $F_0N(\orb{X})$, the subgroup spanned by sheaves supported in dimension zero.
The two Fourier-Mukai functors
\begin{align*}
	\Phi: D(Y) \rightleftarrows D(\orb{X}): \Psi
\end{align*}
induce an isomorphism on the level of numerical K-groups.
\begin{align*}
	\phi: N(Y) \rightleftarrows N(\orb{X}): \psi
\end{align*}
The group $F_0N(\orb{X})$ is sent isomorphically to $F_{\exc}N(Y)$ via $\psi$ and we define $F_{\mr} N(\orb{X})$ to be the image under $\phi$ of $F_1N(Y)$ (the subscript $\mr$ stands for \emph{multi-regular} \cite{crc}).
As it will be useful later, we analogously define $F_0N(Y)$ to be the subgroup spanned by sheaves supported in dimension zero.
\begin{center}
	\begin{tikzpicture}
		\matrix (m) [matrix of math nodes, row sep=1.5em, column sep=.7em, text height=1.5ex, text depth=0.25ex]
		{
		F_0N(Y) & F_{\exc}N(Y) & F_1N(Y) &\\
		& F_0N(\orb{X}) & F_{\mr}N(\orb{X}) & F_1N(\orb{X}) \\
		};
		\path[color=white]
		(m-1-1) edge node[color=black]{$\subset$} (m-1-2)
		(m-1-2) edge node[color=black]{$\subset$} (m-1-3)
				edge [color=black,->] node[auto]{\rotatebox{90}{$\sim$}} (m-2-2)%node [auto] {\rotatebox{90}{$\sim$}} (m-2-2)
		(m-1-3) edge [color=black,->] node[auto]{\rotatebox{90}{$\sim$}} (m-2-3)% node [auto] {\rotatebox{90}{$\sim$}} (m-2-3)
		(m-2-2) edge node[color=black]{$\subset$} (m-2-3)
		(m-2-3) edge node[color=black]{$\subset$} (m-2-4)
		;
	\end{tikzpicture}
\end{center}
Before we may proceed, a technical remark is in order.
\begin{rmk}\label{sign issue crc}
	As mentioned in the introduction, we think of DT invariants as weighted Euler characteristics of the Hilbert scheme of a given Calabi-Yau threefold $M$, where the weight is given by Behrend's constructible function.
	The proof of the flop formula in \cite{cala} relies on the technology of motivic Hall algebras of Joyce.
	One of the technical points of this approach is that the Hilbert scheme $\Hilb(M)$ has a forgetful morphism $\sigma$ to the stack of coherent sheaves $\orb{M}$.
	Thus, on $\Hilb(M)$ there are two candidate constructible functions: the Behrend function $\nu_{\Hilb(M)}$ and the pullback $\mu = \sigma^* \nu_{\orb{M}}$ of the Behrend function on $\orb{M}$.
	Fortunately there is a simple relationship between the two, which unfortunately introduces some signs: namely if $\O_M \onto E$ is a quotient, with $E$ supported in dimension at most one then \cite[Theorem 3.1]{tomcc}
	\begin{align*}
		\nu_{\Hilb(M)}(\O_M \onto E) = (-1)^{\chi(E)} \mu(\O_M \onto E) = (-1)^{\chi(E)} \nu_{\orb{M}} (E)
	\end{align*}
	where $\chi(E) = \chi(\O_M,E)$ is the Euler characteristic of the sheaf $E$.
	Moreover, and this is relevant to the $\pDT$ numbers, if $f\colon U \to Z$ is an open immersion then $\nu_U = f^*\nu_Z$.
\end{rmk}
The \emph{DT number} of $Y$ of class $\alpha \in F_1N(Y)$ is given by 
\begin{align*}
	\DT_Y(\alpha) := \chi_{\topp}\left( \Hilb_Y(\alpha), \nu \right)
\end{align*}
but for convenience, we give a name to the numbers obtained by weighing with $\mu$ as well, namely
\begin{align*}
	\correct{\DT}_Y(\alpha) := \chi_{\topp}\left( \Hilb_Y(\alpha), \mu \right) = (-1)^{\chi(\alpha)} \DT_Y(\alpha).
\end{align*}
In the introduction we also mentioned that we package all these numbers into a generating series
\begin{align*}
	\DT(Y) := \sum_{\alpha \in F_1N(Y)} \DT_Y(\alpha)q^\alpha
\end{align*}
and similarly for $\correct{\DT}$.

Recall now the category $\Per(Y/X)$ of perverse coherent sheaves from the previous section.
The structure sheaf $\O_Y$ belongs to $\Per(Y/X)$ and there is a moduli space $\PHilb(Y/X)$ parameterising quotients of $\O_Y$ in $\Per(Y/X)$ \cite[Section 6]{tomflops}.
This space splits into open and closed components $\PHilb_{Y/X}(\alpha)$, for each numerical class $\alpha$, parameterising quotients $\O_Y \onto P$, with $[P] = \alpha$.
However, there is also an \emph{open} subspace $\PHilb_{\leq 1}(Y/X) \subset \PHilb(Y/X)$ parameterising quotients $\O_Y \onto P$ with $\dim \supp P \leq 1$, and we define $\PHilb^\partial_{Y/X}(\alpha)$ to be the intersection $\PHilb_{\leq 1}(Y/X) \cap \PHilb_{Y/X}(\alpha) \subset \PHilb_{Y/X}(\alpha)$.
A priori, for $\alpha \in F_1N(Y)$, $\PHilb^\partial_{Y/X}(\alpha)$ might differ from $\PHilb_{Y/X}(\alpha)$.

We can define a \emph{perverse} DT number of $Y$ over $X$ of class $\alpha$ as the weighted Euler characteristic
\begin{align*}
	\correct{\DT}_{Y/X}(\alpha) := \chi_{\topp}\left( \PHilb^\partial_{Y/X}(\alpha), \mu \right)
\end{align*}
where $\mu$ is the pullback of the Behrend function of the stack of perverse coherent sheaves on $Y$.
We also collect these numbers into a generating series
\begin{align*}
	\correct{\DT}(Y/X) := \sum_{\alpha \in F_1N(Y)} \correct{\DT}_{Y/X}(\alpha) q^\alpha.
\end{align*}

On the orbifold side, we once again define DT numbers by taking weighted Euler characteristics and gather them in a generating series
\begin{align*}
	\DT_{\mr}(\orb{X}) := \sum_{\alpha \in F_{\mr}N(\orb{X})} \DT_{\orb{X}}(\alpha) q^\alpha,
	\quad \DT_{\orb{X}}(\alpha) := \chi_{\topp}\left( \Hilb_\orb{X}(\alpha), \nu \right).
\end{align*}
\begin{rmk}\label{sign issue again crc}
	The analogue of Remark \ref{sign issue crc} for $\orb{X}$ still holds, that is the following identity holds
	\begin{align*}
		\chi_\text{top} \left( \Hilb_{\orb{X}}(\alpha), \mu \right) = (-1)^{\chi(\alpha)} \chi_\text{top}\left( \Hilb_{\orb{X}}(\alpha), \nu \right)
	\end{align*}
	where $\mu$ is the pullback of the Behrend function of the stack of coherent sheaves on $\orb{X}$.
	To prove this, one can choose an appropriate divisor $D$ on the coarse space $X$, and its pullback to $\orb{X}$ plays the role of $H$ in the proof of \cite[Theorem 3.1]{tomcc}.
	The affine $U$ can then be chosen to be an \'{e}tale open in $\orb{X}$, so that \cite[Lemma 3.2]{tomcc} can be applied.
\end{rmk}
Because of this remark, we can define the underlined version of $\DT_{\mr}(\orb{X})$ and the identity above translates to
\begin{align*}
	\correct{\DT}_{\orb{X}}(\alpha) = (-1)^{\chi(\alpha)}\DT_\orb{X}(\alpha).
\end{align*}

\begin{rmk}\label{quot}
	Form the previous section we know that the Fourier-Mukai equivalences $\Phi$ and $\Psi$ restrict to an equivalence of abelian categories between $\Per(Y/X)$ and $\Coh (\orb{X})$.
	Using Remark \ref{o}, which tells us that $\Phi(\O_Y) = \O_\orb{X}$, we have an induced isomorphism of Quot functors (or Hilbert schemes), hence
	\begin{align*}
		\Hilb_\orb{X}(\alpha) \simeq \PHilb_{Y/X}(\psi(\alpha)).
	\end{align*}
	We define the open subspace $\Hilb^\partial_\orb{X}(\alpha)$ to agree, under this identification, with $\PHilb^\partial_{Y/X}(\psi(\alpha))$.
	By taking weighted Euler characteristics we obtain the analogues $\pDT$, $\underline{\text{DT}}^\partial$ of the generating series for DT invariants.
\end{rmk}

\begin{rmk}\label{technical}
	Before we state the theorem, we point out a technical detail.
	In \cite{tomcc} the generating series $\DT(Y)$ is interpreted as belonging to an algebra $\C[\Delta]_\Phi$ (where $\Delta \subset F_1N(Y)$ is the positive cone of classes $[E]$ with $E \in \Coh(Y)$),  whose elements consist of formal series
	\begin{align*}
		\sum_{(\beta,n) \in \Delta \subset N_1(Y) \oplus \Z} a_{(\beta,n)} q^{(\beta,n)}
	\end{align*}
	where the $a_{(\beta,n)}$ are complex coefficients such that, for a fixed $\beta$, $a_{(\beta,n)}=0$ for $n$ very negative.
	
	A similar interpretation is given in \cite{cala} for the generating series $\correct{\DT}(Y/X)$, which now belongs to an algebra\footnote{Here the subscripts $\Phi$ and $\Lambda$ are just notation and stand for entirely parallel constraints. Also, to be pedantic, in \cite{cala} $\mathbb{Q}$ was used in place of $\mathbb{C}$. However the latter is obtained by the former by tensoring with $\mathbb{C}$.} $\C[\pepe{\Delta}]_\Lambda$ (where $\pepe{\Delta} \subset F_1N(Y)$ is the positive cone of classes $[P]$ with $P \in \Per(Y/X)$), which is the analogous of $\C[\Delta]_\Phi$ for $\Per(Y/X)$.
	It was remarked in \cite[Remark 3.26]{cala} that the generating series $\correct{\DT}(Y)$ can also be seen as an element of $\C[\pepe{\Delta}]_\Lambda$ and all the identities we write down below should be interpreted as taking place within this algebra.
\end{rmk}
In light of \ref{quot} our main theorem is now immediate.
\begin{thmm}\label{main}
	Assume to be working in Situation \ref{situ2}.
	The following formula holds
	\begin{align*}
		\correct{\DT}^\partial_{\mr}(\orb{X}) = \correct{\DT}(Y/X)
	\end{align*}
	after an identification of variables via $\phi$.
	In particular, for each $\alpha \in F_1N(Y)$
	\begin{align*}
			\correct{\DT}_{Y/X}(\alpha) = \correct{\DT}^\partial_\orb{X}(\phi(\alpha)).
	\end{align*}
	% \begin{align*}
	% 	\psi\left(\DT_{\mr}(\orb{X})\right) = \sum_{\alpha \in F_1N(Y)}\DT_{\orb{X}}(\phi(\alpha)) q^\alpha = \DT(Y/X).
	% \end{align*}
\end{thmm}
% \begin{prf}
% 	Form the previous section we know that the Fourier-Mukai equivalences $\Phi$ and $\Psi$ restrict to an equivalence of abelian categories between $\Per(Y/X)$ and $\Coh (\orb{X})$.
% 	Using Remark \ref{o}, which tells us that $\Phi(\O_Y) = \O_\orb{X}$, we have an induced isomorphism of Quot functors (or Hilbert schemes), hence
% 	\begin{align*}
% 		\Hilb_\orb{X}(\alpha) \simeq \PHilb_{Y/X}(\psi(\alpha)).
% 	\end{align*}
% 	By definition $\psi (F_{\mr}N(\orb{X})) = F_1N(Y)$, from which the result for the generating series follows.
% \end{prf}

\begin{cor}\label{mr=exc}
	Assume to be working in Situation \ref{situ2} and recall the identification of variables from the previous theorem.
	The following formula is true
	\begin{align}\label{onceagain}
		\DT^\partial_{\mr}(\orb{X}) = \frac{\DT^\vee_{\exc}(Y)\DT(Y)}{\DT_0(Y)}
	\end{align}
	where
	\begin{align*}
		\DT_0(Y) &:= \sum_{n \in F_0N(Y)}\DT_Y(n)q^n, \\%\quad \text{ and} \\
		\DT_\text{exc}^\vee &:= \sum_{\substack{(\beta,n) \in N_1(Y)\oplus \Z \\ f_*\beta = 0}} \DT_Y{(-\beta,n)} q^{(\beta,n)}.
	\end{align*}
\end{cor}
\begin{prf}
	First we notice that we can get rid of the underlines thanks to Remarks \ref{sign issue crc} and \ref{sign issue again crc}.
	From the previous theorem the statement we wish to prove is equivalent to proving that $\correct{\DT}(Y/X)$ is equal to the right hand side of \eqref{onceagain} (modulo the underlines).
	What prevents us from simply applying the formula \eqref{flopfo} from the introduction is that in \cite{cala} the result is, strictly speaking, only proved for the minus one perversity.
	
	Nonetheless, we can verify that the context we work in here satisfies the hypothesis of \cite[Remark 1.6]{cala}.
	The only thing to prove is that the stack of perverse coherent sheaves is locally isomorphic to the stack of coherent sheaves (via an \emph{exact} functor).
	However, using the Fourier-Mukai equivalence $\otwit{\Phi}$, we have that the stack parameterising objects in $\perv{0}{\Per(Y/X)}$ is isomorphic to the stack parameterising objects in $\perv{-1}{\Per(Y/X)}$.
	As $\otwit{\Phi}$ is also an exact functor, all the constructions of \cite{cala} go through and \eqref{flopfo} does indeed hold.
\end{prf}
Finally, we reformulate our formula paralleling the Crepant Resolution Conjecture of \cite{crc}.
\begin{cor}\label{zero=exc}
	Once again, we assume to be working in Situation \ref{situ2}, while also bearing in mind the identification of variables from the previous theorem.
	The following formulae hold.
	\begin{align}\label{crepe1}
		\DT^\partial_0(\orb{X}) = \frac{\DT_{\exc}^\vee(Y) \DT_{\exc}(Y)}{\DT_0(Y)}
	\end{align}
	\begin{align}\label{crepe2}
		\frac{\DT^\partial_{\mr}(\orb{X})}{\DT^\partial_0(\orb{X})} = \frac{\DT(Y)}{\DT_{\exc}(Y)}
	\end{align}
	where
	\begin{align*}
		\pDT_0(\orb{X}) := \sum_{\alpha \in F_0N(\orb{X})} \pDT_\orb{X}(\alpha) q^\alpha,  \quad\text{ and }\quad
		\DT_{\exc}(Y) := \sum_{\substack{(\beta,n) \in N_1(Y) \oplus \Z \\ f_* \beta = 0}} \DT_Y(\beta,n)q^{(\beta,n)}.
	\end{align*}
\end{cor}
\begin{prf}
	As previously mentioned, $\psi$ identifies $F_0N(\orb{X})$ with $F_{\exc}N(Y)$, from which we deduce the first identity.
	The second is obtained by combining the first identity with \eqref{onceagain}.
\end{prf}
\begin{rmk}\label{remark}
	The actual Conjectures 1 and 2 of \cite{crc} are \eqref{crepe2} and \eqref{crepe1} where the $\partial$ symbols do not appear.
	The similarity between these is so striking that there must be a hidden relationship lurking in the background.
	First off, at the time of writing the author does not know whether or not the inclusion $i\colon \PHilb^\partial_{Y/X}(\alpha) \subset \PHilb_{Y/X}(\alpha)$ is strict.
	If $i$ turned out to be the identity then $\pDT = \DT$ and all formulae match up with \cite{crc}.
	Secondly (and very unlikely) there might be a corner of the universe of hard Lefschetz CY3 orbifolds where the conjectures in \cite{crc} break down and must be replaced.
	The third (and perhaps more likely) hypothesis may be that $i$ is generally a strict inclusion but the two moduli spaces,
	as elements in the Hall algebra of $\Per(Y/X)$, are sent to one another by an automorphism which becomes the identity under the integration map.
	This might be related to the Bryan-Steinberg \emph{relative stable pair} invariants of \cite{david}.
	Steinberg constructs moduli spaces $f\text{-}\Hilb(Y/X)$, which morally parameterise quotients of $\O_Y$ in yet another heart in $D(Y)$.\footnote{We use the word \emph{morally} as when $f$ contracts a divisor to a curve this is no longer true. On the other hand, when $f$ \emph{is} a small resolution then one does get another heart, which turns out to be a tilt of $\Per(Y/X)$. In this (much simpler!) setting the formula for BS invariants can be proved using \cite{cala}.}
	Taking weighted Euler characteristics one has new invariants $\BS(Y/X)$ and the main result of \cite{david} is the formula
	\begin{align*}
		\BS(Y/X) = \frac{\DT(Y)}{\DT_\text{exc}(Y)},
	\end{align*}
	cfr.~\eqref{crepe2}.
	One would hope $\BS(Y/X)$ to match up with the PT invariants of $\orb{X}$, which through an orbifold DT/PT correspondence (announced by Arend Bayer a few years ago) would yield the crepant resolution conjecture.
	Unfortunately, even in the local $\text{B}{\Z_2}$-gerbe case of \cite{crc} there are relative stable pairs which are not sent to stable pairs on the orbifold under the Fourier-Mukai equivalence.
	It is reasonable to suspect that this issue is connected to the appearance of $\partial$ in our formulae.
	At any rate, it would at least be worthwhile to describe the moduli space $\Hilb_\orb{X}^\partial$ intrinsically in terms of $\orb{X}$.
\end{rmk}
% section the_formula_for_dt_invariants (end)












crepresconj.bbl
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
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