Counting surfaces on Calabi-Yau 4-folds I]Counting surfaces on Calabi-Yau 4-folds I: foundations
Y. Bae, M. Kool, H. Park]Younghan Bae, Martijn Kool, and Hyeonjun Park

This is the first part in a series of papers on counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce two types of moduli spaces of stable pairs. We show that all three moduli spaces are related by GIT wall-crossing and parametrize stable objects in the bounded derived category.

We construct  reduced Oh-Thomas virtual cycles on the moduli spaces via Kiem-Li cosection localization and prove that they are deformation invariant along Hodge loci. As an application, we show that the variational Hodge conjecture holds for any family of Calabi-Yau 4-folds supporting a non-zero reduced virtual cycle.

§ INTRODUCTION

§.§ Background

Counting curves on a space $X$ is one of the central topics in modern enumerative geometry. It is a rich subject with links to many other fields of mathematics as well as physics (string theory).
In [100] Kontsevich introduced the concept of hidden smoothness of moduli spaces of stable maps in order to define algebraic Gromov-Witten (GW) invariants of $X$. The mathematical foundation of its virtual cycles was established by Li-Tian [106] and Behrend-Fantechi [12]. When $X$ is a 3-fold, an alternative enumeration can be carried out using Grothendieck's Hilbert schemes. This change of perspective represents the difference between viewing curves on $X$ as being parametrized or cut out by equations. In [147], Thomas then defined virtual cycles on Hilbert schemes of a 3-fold $X$, which yields Donaldson-Thomas (DT) invariants[[147] defines DT invariants for stable sheaves of any rank on Fano and Calabi-Yau $3$-folds.] of $X$. A remarkable connection between GW and DT invariants for 3-folds was discovered by  Maulik-Nekrasov-Okounkov-Pandharipande [117]. This correspondence was further refined by Pandharipande-Thomas via stable pair (PT) invariants [128, 129, 130]. We refer the reader to [131] for an overview.

It was recently found that a similar relation between curve counting invariants exists on Calabi-Yau 4-folds $X$. The structure of GW invariants of $X$ was studied by Klemm and Pandharipande [97]. On the sheaf side, the obstruction theories are not 2-term, and a new discovery was needed. Inspired by work of Donaldson and Thomas [50], Cao and Leung [38] defined such a virtual cycle in special cases. The general case was subsequently established by Borisov and Joyce [23] using methods of derived differential geometry. The algebraic counterpart was obtained by Oh and Thomas [125]. Both constructions [23, 125] deeply rely on the derived structure of the moduli space — the shifted symplectic structures of Pantev-Toën-Vaquié-Vezzosi [132] and the derived Darboux theorem of Brav-Bussi-Joyce [27].
Another feature of the theory is that it requires orientations.
The existence of orientations was shown by Cao-Gross-Joyce [34].
An explicit GW-PT correspondence on Calabi-Yau 4-folds was conjectured by Cao, Maulik and Toda [39, 40, 45]. Virtual counts of points and curves on Calabi-Yau 4-folds recently attracted quite some attention.
In physics, Nekrasov-Piazzalunga's Magnificent Four conjecture [123, 124]
provides a formula for counting points on the Calabi-Yau 4-fold $\C^4$. For the case of counting points on $X$ we refer to [35, 133, 21, 101] and for the case of counting curves on $X$ we refer to [36, 37, 43, 44, 133].

The next natural step in enumerative geometry is counting higher-dimensional objects, such as surfaces on Calabi-Yau 4-folds — the topic of this paper. It is not clear at present how to define virtual cycles on the stable map side (e.g., see Alexeev [2] for the moduli space and Jiang [87] for some recent developments).
On the sheaf side, a natural starting point is the virtual cycle of [23, 125] on Hilbert schemes of 2-dimensional subschemes. However, we will see that for enumeration of surfaces, this virtual cycle generally does not suffice and has to be modified.

§.§ Counting surfaces

The aim of this project is to count surfaces on Calabi-Yau 4-folds $X$ via sheaf theory. This goal leads to two immediate challenges.

(A) The Hilbert scheme parametrizes subschemes which are not pure dimensional, i.e. 2-dimensional subschemes may have 0- and 1-dimensional components. Informally: Hilbert schemes allow for free roaming curves and points thereby obscuring the contribution of genuine surfaces.

(B) A Hodge class of type $(2,2)$ on $X$ generally does not remain of type $(2,2)$ as the complex structure of $X$ deforms. When this happens, the virtual cycle vanishes.

The main purpose of this paper is to resolve (A) and (B).

Inspired by Pandharipande-Thomas's work on curves on 3-folds [128], we settle (A) by constructing moduli spaces of stable pairs on $X$. We find that for 2-dimensional pairs, there is more structure than for 1-dimensional pairs, and we obtain two types which we call $\PT_0$ and $\PT_1$ pairs. The stable pair moduli spaces give smaller compactifications — roughly speaking, $\PT_0$ pairs prevent free roaming points on $X$ and $\PT_1$ pairs prevent free roaming points and curves on $X$. The Hilbert scheme and two stable pair moduli spaces each have desirable features and should be studied together.

We overcome (B) by constructing reduced virtual cycles using the cosection localization method of Kiem and Li [94].
This method was generalized to the case of isotropic cosections by Kiem and the third-named author [96].
Instead in this paper, we use a variant, namely non-degenerate cosections.
For a class $\gamma\in H^{2}(X,\Omega_X^2)$, virtual cycles of moduli spaces parametrizing 2-dimensional pairs in class $\gamma$ generally vanish.
Similar vanishing for curve counting theories on surfaces/3-folds led to reduced theories e.g. [118, 102] (we refer the reader to the introductions of these papers for more references). From the perspective of curves on surfaces/3-folds $X$, reducing is often needed when $X$ has a holomorphic 2-form. Therefore, for compact Calabi-Yau 3-folds satisfying $h^{0,1}(X) = h^{0,2}(X) = 0$, it is of no concern. As we will see, even for surfaces on compact Calabi-Yau 4-folds satisfying $h^{0,1}(X) = h^{0,2}(X) = h^{0,3}(X) = 0$ reducing is almost always necessary.

§.§ Moduli spaces of stable pairs

There are three types of stability conditions on $2$-dimensional pairs.

Let $X$ be a smooth projective variety of dimension $n\geq 2$.
Let $F$ be a 2-dimensional coherent sheaf on $X$, $s:\O_X\to F$ a section, and $Q:=\coker(s)$. The pair $(F,s)$ is called

(i) $\DT(=\PT_{-1})$-stable if $s$ is surjective,

(ii) $\PT_0$-stable if $T_0(F)=0$ and $\dim(Q)\leq 0$,

(iii) $\PT_1$-stable if $T_1(F)=0$ and $\dim(Q)\leq 1$.

Here $T_d(F)$ denotes the maximal $d$-dimensional subsheaf of $F$.

The $\DT$-stable pairs correspond to ideal sheaves.
The $\PT_1$-stable pairs are natural generalizations of Pandharipande-Thomas's stable pairs [128], which are a special case of Le Potier's coherent systems [135, 136].
The underlying sheaf of a $\PT_0$-stable pair is usually not pure and this stability condition appears to be new.
We introduce $\PT_0$-stability as a bridge between $\DT$- and $\PT_1$-stability.

Suppose $(F,s)$ is a $\PT_q$ pair on $X$ with cokernel $Q$ and smooth scheme-theoretic support $S$, which generally has 0-, 1-, and 2-dimensional connected components.
For $q=0$, $S$ has no 0-dimensional connected components and $Q$ is supported on the 1-dimensional connected components of $S$. For $q=1$, $S$ has no 0- or 1-dimensional components and the pair $(F,s)$ can be equivalently described by a nested triple $Z \subset C \subset S$, where $C$ is an effective divisor and $Z$ is a 0-dimensional subscheme (Corollary <ref>). We give various characterizations of $\PT_q$ pairs with singular support in Section <ref>.

[Proposition <ref>]
Suppose $(F,s)$ is a pair which is $\PT_0$- and $\PT_1$-stable, i.e., $F$ is pure and $Q$ is 0-dimensional.
Then $Q$ is a quotient sheaf of the 0-dimensional sheaf $\ext^{n-1}_X(\O_S,K_X)$, which is supported on the locus where $S$ is not Cohen-Macaulay.

Our first main result (Section <ref>) is the construction of projective moduli spaces  of $\PT_q$-stable pairs via geometric invariant theory (GIT).

Let $X$ be a smooth projective variety and $v\in H^*(X,\mathbb{Q})$. Then, for each $q \in \{-1,0,1\}$, there exists a fine moduli space $\PTqvX$ of $\PT_q$-stable pairs on $X$ with $\ch(F)=v$ as a projective scheme.
Moreover, there exists a projective scheme $\cM$, an action of $\PGL_N$ on $\cM$, and an $\SL_N$-equivariant ample line bundle $\cL^{(q)} \in \Pic^{\SL_N}(\cM)_{\Q}$, for each $q \in \{-1,0,1\}$, such that
\[\curP^{(q)}_v(X) = [\cM^{st}(\cL^{(q)})/\PGL_N] = [\cM^{ss}(\cL^{(q)})/\PGL_N].\]

In particular, $\PTqvX$ is the GIT quotient $ \cM/\!/_{\cL^{(q)}}\SL_N$.

Theorem <ref> shows that $\PT_q$ pair moduli spaces are related by GIT wall-crossing.
Since the underlying sheaves of $\PT_0$ pairs are not pure, we needed finer arguments than the standard ones for pure sheaves.
This approach is inspired by an analogous result of Stoppa-Thomas on 1-dimensional pairs [146].

Following [128], we show that the moduli space of $\PT_q$ pairs consists of open components of the moduli space of perfect complexes. Denote by $\cPerf(X)_{\O_X}^{\mathrm{spl}}$ the moduli space of simple perfect complexes on $X$ with trivial determinant.

Let $X$ be a smooth projective variety of dimension $n\geq 4$, $v \in H^*(X,\Q)$, and  $q\in \{-1,0,1\}$.
The canonical map

\begin{equation}\label{eq:1.1}
\curP^{(q)}_v(X) \to \cPerf(X)_{\O_X}^{\mathrm{spl}}, \quad (F,s) \mapsto I\udot=[\O_X \xrightarrow{s} F]
\end{equation}

is an open embedding.
Moreover, there is a polynomial Bridgeland stability condition $\rho^{(q)}$ on $D^b_{\mathrm{coh}}(X)$
such that the image of the map (<ref>) is the locus of $\rho^{(q)}$-stable objects.

For $q=1$, Theorem <ref> was also shown by Gholampour-Jiang-Lo [59].
The connection to Bayer stability is inspired by [59].

§.§ Reduced virtual cycles

Let $X$ be a Calabi-Yau $4$-fold, i.e., a smooth projective variety of dimension $4$ with trivial canonical bundle. In particular, $X$ could be a hyperkähler $4$-fold or an abelian $4$-fold.
Fix a trivialization $\omega : \O_X \xrightarrow{\cong}\Omega^4_X$.

Throughout this section, let $q \in \{-1,0,1\}$ and let $\curP := \PTqvX$ be the moduli space of $\PT_q$ pairs $(F,s)$ on $X$ with Chern character

\[\ch(F)=v=(0,0,\gamma,\beta,n-\gamma\cdot\td_2(X)) \in H^*(X,\Q).\]

By Theorem <ref>,  we have an Oh-Thomas virtual cycle [125]

\[\left[\curP\right]^\vir \in A_{\vd}\left(\curP\right), \qquad \vd = n - \frac12 \gamma^2,\]

which maps to Borisov-Joyce's virtual cycle [23] under the cycle class map by [126].[Here $A_*(-)$ denotes the Chow group with rational coefficients. We also use the convention that $A_{i}(-)=0$ for $i \in \Q\setminus\Z$.]
The Oh-Thomas virtual cycle is constructed from the
canonical 3-term symmetric obstruction theory

\[\phi : \EE:=R\hom_{\pi}(\II\udot,\II\udot)_0[3] \to \LL_{\curP}, \qquad \SD : \EE\dual[2] \cong \EE,\]

of Huybrechts-Thomas [81] and
an orientation

\[ o  : \O_{\curP} \xrightarrow{\cong} \det(\EE)\]

of Cao-Gross-Joyce  [34].
Then the intrinsic normal cone is isotropic by the derived Darboux theorem of Brav-Bussi-Joyce [27], which is necessary to define the virtual cycle. We refer to Section <ref> for the precise statements.

For a generic surface class $\gamma$, the Oh-Thomas virtual cycle $[\curP]^\vir$ vanishes due to the following two heuristics:

(a) When deforming $X$, the codimension of the Hodge locus of $\gamma$ in the base is generally positive.
Therefore deformation invariance of virtual cycles implies the vanishing.

(b) In general, there exist non-degenerate cosections given by semi-regularity maps.
They give trivial factors $\O$ of $\EE$ which imply the vanishing.

We present a motivating example which reveals the above two phenomena.

Let $X \subseteq \PP^5$ be a smooth sextic hypersurface which contains a plane $S\subseteq X$. Let $v=\ch(\O_S) \in H^*(X,\Q)$ and $q\in\{-1,0,1\}$.

* The virtual dimension is $-\frac{19}{2}$.

* The codimension of the Hodge locus of $\gamma=v_2=[S]$ is $19$.

* There are $19$ non-degenerate cosections.

We first discuss the codimension of the Hodge locus in (a).
By Bloch [19], the obstruction to deforming the Hodge class $\gamma \in H^2(X,\Omega^2_X) \subseteq H^4(X,\C)$ is governed by the symmetric bilinear form

\begin{equation*}\label{eq:1.2}
\mathsf{B}_\gamma : H^1(X,T_X) \otimes H^1(X,T_X) \to \C, \quad \xi_1 \otimes \xi_2 \mapsto \int_X \iota_{\xi_1}\iota_{\xi_2}\gamma \cup \omega\,,
\end{equation*}

where $\iota$ denotes the contraction map. In particular, its rank

\begin{equation*}\label{eq:1.2.1}
\rho_\gamma :=\rk (\mathsf{B}_\gamma)
\end{equation*}

is the codimension of the Hodge locus of $\gamma$.[Let $f:\cX\to\cB$ be a smooth proper morphism of smooth analytic spaces such that the fibre $\cX_0$ over a point $0 \in \cB$ is $X^\an$ and $\cB$ is contractible. Then there exists a unique lift $\tgamma$ of $\gamma \in H^4(X,\Q)$ as a section of the local system $R^4f_* \Q$. The Hodge locus of $\gamma$ is defined as the subset $\mathrm{Hd}_\gamma:=\{b\in\cB : \tgamma_b \in H^{2,2}(\cX_b)\}$ and has a natural analytic scheme structure (see [160]). If the Kodaira-Spencer map $\KS : T_{\cB,0} \to H^1(X,T_X)$ is an isomorphism, then $\rho_\gamma = \dim(T_{\cB,0}) -\dim(T_{\mathrm{Hd}_\gamma,0})$ by [19].
We will not use the Hodge locus in this paper except for heuristic arguments.]

We now discuss the cosections in (b).
By Buchweitz-Flenner [30], a perfect complex $[I\udot]\in \curP$ gives rise to a semi-regularity map

\begin{equation}\label{eq:sr1}
\sr : \Ext^2_X(I\udot,I\udot)_0 \to H^3(X,\Omega_X^1)\cong H^1(X,T_X)\dual, \quad e\mapsto \tr(\At(I\udot)\circ e),
\end{equation}

where $\At(I\udot)$ is the Atiyah class.
This semi-regularity map relates the obstructions of deforming the perfect complex $I\udot$ and the Hodge class $\gamma=\ch_2(I\udot)$ via the commutative triangle

\begin{equation}\label{eq:t1}
\xymatrix{ H^1(X,T_X) \ar[rr]^-{\sfob=\sr\dual} \ar[rd]_-{\mathsf{B}_\gamma} && \Ext^2_{X}(I\udot,I\udot)_0  \ar[ld]^-{\sr} \\ & H^1(X,T_X)\dual}.
\end{equation}

The symmetry of the above triangle given by $\sfob=\sr\dual$ is a special feature of Calabi-Yau 4-folds $X$ and $(2,2)$-classes $\gamma \in H^2(X,\Omega^2_X)$.
The semi-regularity maps $\sr$ can be extended to cosections on the moduli space,

\[\SR : \EE\dual[1]=R\hom_{\pi}(\II\udot,\II\udot)_0[2] \to H^1(X,T_X)\dual \otimes \O_{\curP}.\]

Then $\SR^2 := \SR\circ\SR\dual = \sfB_\gamma\otimes 1_{\curP}$, which can be shown by extending the triangle (<ref>). See section <ref> for details.

Choose a non-degenerate subspace $V \subseteq H^1(X,T_X)$ with respect to the bilinear form $\sfB_\gamma$. Then $V \cong V^\vee$.
Assume that $V$ is maximal, i.e., $\dim(V)=\rho_\gamma$.

(a') From the perspective of Hodge theory, $V$ is a transversal slice to the Hodge locus of $\gamma$. Heuristically, we want to constrain virtual cycles to be deformation invariant only on the Hodge locus of $\gamma$.

(b') From the perspective of cosections, $V$ induces non-degenerate cosections

\[\Sigma : \EE\dual[1] \xrightarrow{\SR} H^1(X,T_X)\dual \otimes \O_{\curP} \to V\dual \otimes \O_{\curP}.\]

Here non-degeneracy means that the square $\Sigma^2:=\Sigma\circ\Sigma\dual$ is an isomorphism.
In particular, we have a direct sum decomposition

\[\EE = \EE^\red \oplus (V\otimes \O_{\curP})[1]\]

and we want to remove the trivial factors from the obstruction theory.

The second main result in this paper achieves (a') and (b').

Let $X$ be a Calabi-Yau 4-fold, $v \in H^*(X,\Q)$, and $q\in \{-1,0,1\}$. For any maximal non-degenerate subspace $V \subseteq H^1(X,T_X)$ with respect to $\sfB_{\gamma}$, the composition

\[\phi^\red :\EE^\red \xrightarrow{(1,0)} \EE \xrightarrow{\phi} \LL_{\PTqvX}\]

is a symmetric obstruction theory such that the reduced closed substack of the intrinsic normal cone is isotropic (see Section <ref> for the precise definition).
Consequently, given orientations of $\EE$ and $H^1(X,T_X)_\gamma:=H^1(X,T_X)/\ker(\sfB_\gamma)$, there exists a canonical reduced virtual cycle

\[\left[\PTqvX\right]^\red \in A_{\rvd}\left(\PTqvX\right), \qquad \rvd = n- \frac12 \gamma^2 + \frac12 \rho_\gamma,\]

which is independent of the choice of $V$.

We prove that $\phi^\red$ is an obstruction theory via the algebraic twistor family [102] associated to the transversal slice $V$ and prove the isotropic condition via a variation of the cosection localization in [94, 96] for the non-degenerate cosections $\Sigma$. In contrast to the isotropic cosections in [96], the non-degenerate cosections considered in this paper allow us to handle cases where the virtual dimension $n-\frac12\gamma^2$ is not an integer, but the reduced virtual dimension $n-\frac12\gamma^2+\frac12\rho_\gamma$ is an integer, which actually occurs quite often (e.g. Example <ref>).

We note that the algebraic twistor method alone does not produce a reduced virtual cycle
since the isotropic condition for $\phi$ does not imply it for $\phi^\red$.
The cosection localization method alone produces the reduced virtual cycle as a short-cut, but does not give a reduced obstruction theory.
Combining the two methods enables us to directly apply all the known tools for handling Oh-Thomas virtual cycles (torus localization [125], cosection localization [96], virtual pullback [133]) to the reduced virtual cycle.
We will indeed apply these tools in the sequels to this paper.

We observe that the reduced virtual cycle is closely related to semi-regularity.
In [30], a perfect complex $[I\udot] \in \curP$ is called semi-regular if the semi-regularity map $\sr$ in (<ref>) is injective.

Let $X$ be a Calabi-Yau 4-fold, $v \in H^*(X,\Q)$, and $q\in \{-1,0,1\}$.
For any $\PT_q$ pair $[(F,s)] \in \PTqvX$, the associated complex $I\udot$ is semi-regular if and only if the moduli space $\PTqvX$ is smooth of dimension $\rvd=n-\frac12\gamma^2+\frac12\rho_\gamma$ at  $[I\udot]$.

As a direct corollary of Theorem <ref>, if all $[(F,s)] \in \PTqvX$ are semi-regular, the reduced virtual cycle equals the fundamental cycle (for a certain choice of orientations).
Thus, heuristically, we view the semi-regular case as the ideal situation. Put differently, we view the reduced virtual cycle as the correct virtual generalization of the fundamental cycle in the semi-regular case.

For Calabi-Yau 4-folds, Theorem <ref> generalizes the results of Bloch [19] and Buchweitz-Flenner [30]. Indeed, it is shown in loc. cit. that semi-regularity implies smoothness, whereas Theorem <ref> also provides a formula for the dimension and a converse.

We revisit  Example <ref>.
All planes $S \cong \PP^2$ in a smooth sextic $4$-fold $X$ are semi-regular and the moduli space ${\curly P}^{(-1)}_v(X) \cong {\curly P}^{(0)}_v(X) \cong {\curly P}^{(1)}_v(X)$, with $v = \ch(\O_S)$, consists of isolated reduced points (Corollary <ref>).
Hence the reduced virtual dimension is zero and the contribution of any plane $S\subset X$ to the reduced virtual cycle  is $\pm[\{S\}]$.

More generally, we give a simple description of the semi-regularity map for local complete intersection surfaces. This identification is important for computing $\rho_\gamma$ and the reduced virtual dimension.
Determining $\rho_\gamma$ is an interesting new aspect of the surface counting theory. For example, $\rho_\gamma$ is only sub-additive in $\gamma$, e.g., $\rho_{d \gamma} = \rho_{\gamma}$ for any $d \in \Z_{>0}$.

Let $i: S\hookrightarrow X$ be a local complete intersection surface inside a Calabi-Yau 4-fold $X$. Then there exists an isomorphism

\[\theta_S: H^1(S,N_{S/X})\to \Ext^2_X(I_{S/X},I_{S/X})_0\]

preserving non-degenerate symmetric bilinear forms induced by Serre duality. Under this isomorphism, the sheaf theoretic obstruction and semi-regularity maps (<ref>) match with the ones from the deformation  theory of subschemes.

We identify the dual of the semi-regularity map (<ref>) with the obstruction map $\ob_{S/X}:H^1(X,T_X)\to H^1(S,N_{S/X})$ for Hilbert schemes, which is easier to determine. This leads to the following corollary:

$(1)$ Let $X\subset \PP^N$ be a smooth complete intersection Calabi-Yau 4-fold. Suppose $S\subset X$ is a complete intersection surface in $\PP^N$. Then $I_{S/X}$ is semi-regular. Moreover if $v=\ch(\O_S)$ and $q \in \{-1,0,1\}$, then $\PTqvX$ is smooth of dimension $\rvd=h^0(N_{S/X})$ at $I_{S/X}$.

$(2)$ Let $X$ be a Calabi-Yau 4-fold with non-degenerate holomorphic $2$-form. Suppose $i:S\hookrightarrow X$ is a smooth lagrangian surface and $v=\ch(\O_S)$. Then $\rvd=h^{1,0}(S) - \frac{1}{2} \dim \coker(i^*)$ where $i^*: H^1(X,\Omega_X^1)\to H^1(S,\Omega_S^1)$.

For Corollary <ref> (1) and $X$ a sextic 4-fold, we show that $\rvd$ is often zero, but can be positive in some cases.
For Corollary <ref> (2), we find that lagrangian planes are semi-regular with $\rvd = 0$, but we also consider Schoen surfaces [140] embedded in their Albanese 4-fold which are semi-regular with $\rvd = 4$.
Furthermore, for Corollary <ref> (2), we also consider non-semi-regular cases on hyperkähler 4-folds with lagrangian fibration and the Hilbert square $K3^{[2]}$.
Finally, we show that $\rho_\gamma$ vanishes when  $h^{2,0}(X)=0$ and the class $\gamma$ is a product of two $(1,1)$-classes. In this case, non-reduced virtual cycles contain non-trivial information.
See Section <ref> for a precise list of examples.

§.§ Deformation invariance

\[f:\cX\to \cB\]

be a family of Calabi-Yau 4-folds, i.e., a smooth projective morphism of relative dimension $4$ to a smooth connected affine scheme $\cB$ such that $\omega_{\cX/\cB} \cong \O_{\cX}$ and with connected fibres. Consider a horizontal section

\[\tv\in \bigoplus_{p=2}^4 F^p H^{2p}_{DR}(\cX/\cB),\]

where $F^p H^{2p}_{DR}(\cX/\cB)$ is the $p$th filtered piece of the Hodge bundle $H^{2p}_{DR}(\cX/\cB)$.
Then $\tv_b \in H^*(\cX_b,\C)$ is a locally constant family of cohomology classes on the fibres $\cX_b:=\cX\times_{\cB}\{b\}$ over $b\in \cB$.
The component $\tv_2$ induces a section  $\tgamma$ of $R^2f_*\Omega^2_{\cX/\cB}$.
The bilinear forms $\sfB_{\tgamma_b}$ on $H^1(\cX_b,T_{\cX_b})$ for $b \in \cB$ extend to a bilinear form $\sfB_{\tgamma}$ on the vector bundle $R^1f_*(T_{\cX/\cB})$.

Let $\PTqvXB$ be the moduli space of $\PT_q$-stable pairs on the fibres of $f:\cX\to\cB$ for some $q \in \{-1,0,1\}$. This relative moduli space fits into the fibre diagram

\begin{equation*}
\xymatrix{
\PTqvXb \ar@{^{(}->}[r]^{} \ar[d] & \PTqvXB \ar[d]^{} \\
\{b\} \ar@{^{(}->}[r]^{i_b} & \cB
\end{equation*}

for each $b\in \cB$. We have a relative obstruction theory $\phi_{\cX/\cB} :\EE_{\cX/\cB} \to \LL_{\PTqvXB/\cB}$.

Our third main result (Section <ref>) is the deformation invariance of the reduced virtual cycle in Theorem <ref>.

Let $f:\cX\to\cB$ be a family of Calabi-Yau 4-folds.
Let $\tv \in \bigoplus_p F^pH^{2p}_{DR}(\cX/\cB)$ be a horizontal section and $\tgamma \in \Gamma(\cB, R^2f_*\Omega^2_{\cX/\cB})$ be the induced section.
Assume that the function $b\in \cB \mapsto \rho_{\tgamma_b}$ is constant and there exist orientations

\[\O_{\PTqvXB} \cong \det(\EE_{\cX/\cB}),\qquad \O_{\cB} \cong \det(R^1f_*(T_{\cX/\cB})/\ker(\sfB_{\tgamma})).\]

Then there exists a cycle class

\[\left[\PTqvXB\right]^\red \in A_{\rvd+\dim(\cB)}\left(\PTqvXB\right)\]

such that, for all $b \in \cB$, we have

\[\left[\PTqvXb\right]^\red = i_b^!\left[\PTqvXB\right]^\red \in A_{\rvd}\left(\PTqvXb\right)\]

for the induced orientation.
Here $i_b^!$ denotes the refined Gysin pullback.

In general, reduced virtual cycles given by cosection localization [94] are deformation invariant only under an additional assumption. In Theorem <ref>, this assumption follows from the fact that the Hodge locus of $\tgamma$ in $\cB$ is $\cB$ itself.

The existence of a relative orientation is required in Theorem <ref>. We conjecture (Conjecture <ref>) that a relative orientation exists after a finite étale base change of the base $\cB$ and we prove its topological counterpart (Corollary <ref>). The existence of a relative orientation is crucial for deformation invariance of numerical invariants, but is not needed in the main application below.

§.§.§ Variational Hodge Conjecture

The Hodge conjecture predicts that for any smooth projective variety $X$, all rational $(p,p)$-classes on $X$ are algebraic.
In [69] Grothendieck introduced a variant of the Hodge conjecture.

Let $X$ be a smooth projective variety and $\gamma$ an algebraic $(p,p)$-class on $X$. For any smooth projective morphism $f:\cX\to \cB$ to a smooth connected scheme $\cB$ and a horizontal section $\tv_p$ of $F^p\cH^{2p}_{DR}(\cX/\cB)$
such that $\cX_0 \cong X$ and $(\tv_p)_0=\gamma$ for some closed point $0 \in \cB$, the cohomology classes $(\tv_p)_b$ are algebraic for all closed points $b\in\cB$.

Our fourth main result is an affirmative answer to Conjecture <ref> when there exists a non-zero reduced virtual cycle.

Let $X$ be a Calabi-Yau 4-fold and let $\gamma$ be a $(2,2)$-class on $X$.
If for some $v \in H^*(X,\Q)$ with $v_2=\gamma$, $q \in \{-1,0,1\}$, and choice of orientation on $\PTqvX$ we have

\[[\PTqvX]^\red \neq 0 \in A_{\rvd}(\PTqvX)\]

then Conjecture <ref> holds for $X$ and $\gamma$.

We prove Theorem <ref> via the deformation invariance in Theorem <ref>. Here we do not require the two technical assumptions in Theorem <ref>.
The function $b\in\cB\mapsto \rho_{\tgamma_b}\in \Z$ is lower semi-continuous and the reduced virtual cycle vanishes if $\rho_{\tgamma_b}$ drops at $0 \in \cB$  (Proposition <ref>).
The assumption on orientations can be removed by working with virtual cycles instead of invariants.

We view Theorem <ref> as a virtual generalization of a result of Buchweitz-Flenner [30] (see also [19]) for Calabi-Yau 4-folds. In fact, [30] implies that Conjecture <ref> holds if there exists a semi-regular $\PT_q$ pair $I\udot=[\O_X\to F]$ on $X$ with $\ch_2(F) = \gamma$.
Theorem <ref> recovers the result of [30] for Calabi-Yau 4-folds, because the reduced virtual cycle equals the fundamental cycle near a semi-regular point (Theorem <ref>).

This application to the variational Hodge conjecture shows that it can be useful to work with the virtual cycle instead of invariants, because one can restrict to each connected component of the moduli space.

[Example <ref>]
We give a non-semi-regular application of Theorem <ref>. Let $X=S_1\times S_2$ be a product of two K3 surfaces. Let $\beta_1\in H^{1,1}(S_1)$, $\beta_2\in H^{1,1}(S_2)$ be two irreducible effective classes with $\beta_1^2\geq 2$ and $\beta_2^2=0$.
Let $\gamma=\beta_1\cup \beta_2$.
Then, for any $q \in \{-1,0,1\}$, we have $\PTqvX=|\O_{S_1}(\beta_1)|\times \PP^1$ for some $v\in H^*(X,\mathbb{Q})$ with $v_2=\gamma$. In this case, the reduced virtual cycle is $\pm h^0(\O_{S_1}(\beta_1))\cdot[\pt\times\PP^1]$. Therefore Conjecture <ref> holds for such $\gamma$.
This is a non-semi-regular case since the reduced virtual cycle differs from the fundamental cycle.

§.§ Moduli stack of semi-stable sheaves

In Appendix <ref>, we generalize the construction of reduced virtual cycles to Artin stacks (which are global quotient stacks). In particular, we consider the moduli stack $\cM^{H,ss}_v(X)$ of Gieseker semi-stable sheaves on $X$ (with respect to a polarization $H$) with Chern character $v \in H^*(X,\Q)$, rigidified by the action of $B\GG_m$.
The Chern character $v$ is arbitrary so this includes moduli of $2$-dimensional torsion sheaves and moduli of higher rank sheaves.

Let $(X,H)$ be a polarized Calabi-Yau 4-fold and $v \in H^*(X,\Q)$.
For any choice of orientation on $\cM^{H,ss}_v(X)$, there exists a canonical reduced virtual cycle

\[[\cM^{H,ss}_v(X)]^\red \in A_{1- \frac12\chi(v,v)+\frac12\rho_\gamma}(\cM^{H,ss}_v(X))\]

where $\gamma:=v_2$ and $\chi(v,v):=\int_Xv\dual \cdot v \cdot \td(X)$.
Moreover, if $[\cM^{H,ss}_v(X)]^\red \neq 0$, then Conjecture <ref> holds for $X$ and $\gamma$.

We construct the reduced virtual cycle as a class in Kresch's Chow group [103] by extending Oh-Thomas's construction [125] and Kiem-Li's cosection localization [94] to Artin stacks (which are global quotients).
This virtual cycle does not provide numerical invariants since there is no push-forward map for $\cM^{H,ss}_v(X) \to \Spec(\C)$ in general. Nonetheless, this virtual cycle is sufficient for applications to the variational Hodge conjecture.

There is a “fixed determinant version” of Theorem <ref> under the assumption $v_1=0$ (Theorem <ref>).

§.§ Derived algebraic geometry

In Appendix <ref>, we revisit Theorem <ref> via derived algebraic geometry.
The main result is the following:

In the situation of Theorem <ref>, there exists a $(-2)$-shifted symplectic derived enhancement of $\PTqvX$ inducing the reduced symmetric obstruction theory $\phi^\red$.

We first introduce general reduction methods via $(-1)$-shifted closed/exact 1-forms (Theorem <ref>).
In order to apply this to the moduli space $\PTqvX$, we show that the semi-regularity maps can be enhanced to $(-1)$-shifted closed $1$-forms (Proposition <ref>).
This is achieved by generalizing the integration map of [132] (Theorem <ref>).

As a byproduct, we obtain a new proof of Bloch's conjecture for surfaces on Calabi-Yau $4$-folds (Remark <ref>). The general case of Bloch's conjecture was shown by Pridham [137] and Bandiera-Lepri-Manetti [10] (and by Iacono-Manetti [84] when the normal bundle is extendable).

§.§ Relations to physics

Finding connections between surface counting on Calabi-Yau 4-folds and string theory is an interesting direction. We briefly point out relations to two (disjoint) parts of the physics literature:

* In M-/F-theory, the complex moduli of a Calabi-Yau 4-fold $X$ are fixed by fixing a rational 4-cycle class called the $G_4$-flux.
Then Gukov-Vafa-Witten [71] introduced an associated superpotential giving rise to a symmetric bilinear form which (up to change of coordinates) coincides with $\sfB_\gamma$. Explicit bounds on the rank $\rho_\gamma$ of $\sfB_\gamma$ appear to be of interest to physicists [25, 26, 13, 63, 112].[We thank Grimm for pointing us to the physics literature, specifically [25].]

* In the series of papers starting from [122], Nekrasov studied the moduli space of (non-commutative) instantons on origami spacetimes in $\mathbb{C}^4$. Our choice of $K$-theoretic insertions for toric Calabi-Yau 4-folds in the sequels (and the choice in [37, 101]) are motivated by Nekrasov-Piazzalunga's study of SUSY Yang-Mills theory on $\C^4$ [123, 124].

§.§ Sequels

We briefly mention the content of the sequels to this paper:

* In Part II we study the $\DT/\PT_0$ correspondence. We conjecture a general correspondence and provide evidence on toric Calabi-Yau 4-folds via a vertex formalism, and on compact Calabi-Yau 4-folds via a virtual Lefschetz principle.

* In Part III we study the $\PT_0/\PT_1$ correspondence in special cases. We consider virtual projective bundles and Weierstrass 4-folds.

* In Part IV we compute $\PT_1$ invariants in some examples including local surfaces and Calabi-Yau 4-folds with a nondegenerate holomorphic 2-form.

We will also further explore the link between the variational Hodge conjecture and the reduced virtual cycles.

§.§ Related work

Reduced virtual cycles for curves on hyperkähler $4$-folds were studied by Cao-Oberdieck-Toda in [41, 42].
For the moduli space of $\PT_1$-pairs on $X$, Theorem <ref> was independently proven by Gholampour-Jiang-Lo [59].
Correspondences for $\PT_q$ invariants in the context of Joyce's wall-crossing framework [65, 88, 89] will be an interesting direction to study.

There are two types of generalizations of cosection localization [94] to Donaldson-Thomas theory of Calabi-Yau $4$-folds [23, 125]. In [139], Savvas showed vanishing of Borisov-Joyce virtual cycles with a nowhere vanishing isotropic cosection.
In [96], Kiem and the third-named author studied cosection localization for Oh-Thomas virtual cycles [125].

§.§ Acknowledgements

We thank Rahul Pandharipande for encouraging us to study surfaces on Calabi-Yau 4-folds, which prompted this project.
The second-named author wishes to thank Richard Thomas for previous collaborations on reduced virtual cycles for curves on surfaces, which obviously impacted this paper. He also thanks Yalong Cao for their collaborations which led him to study Calabi-Yau 4-folds in the first place.
The third-named author thanks Young-Hoon Kiem for teaching him the cosection localization and a previous collaboration on its generalization to Calabi-Yau $4$-folds.
We are also grateful to Markus Upmeier for conversations on orientation problems. We thank Dhyan Aranha, Arkadij Bojko, Amin Gholampour, Thomas Grimm, Yunfeng Jiang, Dominic Joyce, Woonam Lim, Jason Lo, Sergej Monavari, Georg Oberdieck, Jeongseok Oh, Jørgen Rennemo, and Cumrun Vafa for many interesting conversations.
Y.B. is supported by ERC Grant ERC-2017-AdG-786580-MACI and Korea Foundation for Advanced Studies (KFAS).
The project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation
program (grant agreement No. 786580).
M.K. is supported by NWO grant VI.Vidi.192.012.

§.§ Notation and conventions

We use the following notation and conventions throughout the paper.

* All schemes and algebraic stacks are assumed to be of finite type over the field $\C$, unless stated otherwise.
A variety is a separated integral scheme.
A point $x\in X$ in a scheme or an algebraic stack $X$ means a $\C$-valued point.

* A Calabi-Yau $n$-fold is a smooth projective variety $X$ of dimension $n$ with $K_X \cong \O_X$.

* For a coherent sheaf $F$ on a smooth quasi-projective variety $X$, we use the following notation for the various duals:

* $F^\vee := R\hom_X(F,\O_X)$;

* $F^D:=\ext^c_X(F,K_X)$, where $c$ is the codimension of $F$ in $X$.

* For a smooth projective variety $X$, we denote by

* $\Coh(X)$ the abelian category of coherent sheaves on $X$;

* $\cCoh(X)$ the moduli stack of coherent sheaves on $X$;

* $D^b_{\mathrm{coh}}(X)$ the bounded derived category of coherent sheaves on $X$.

* For a perfect complex $\FF$ on a scheme $\sX$ of tor-amplitude $(-\infty,0]$, denote by $\fC(\FF):=h^1/h^0(\FF\dual)$ the associated abelian cone stack <cit.>.

* An orthogonal bundle on a scheme $\sX$ is a pair of a vector bundle $E$ and an isomorphism $q: E\to E\dual$ such that $q=q\dual$. By abuse of notation, we also denote by $q:E \to \bbA^1_{\sX}$ the corresponding quadratic function. An orientation of an orthogonal bundle $E$ is an isomorphism $o : \O_{\sX} \cong \det(E)$ such that $\det(q) = (-1)^{\frac{r(r-1)}{2}}(o \circ o\dual)^{-1}$ where $r=\rank(E)$.

* A symmetric complex on a scheme $\sX$ is a pair of a perfect complex $\EE$ on $\sX$ of tor-amplitude $[-2,0]$ and an isomorphism $\theta: \EE\dual[2] \cong \EE$ such that $\theta\dual[2]=\theta$.  An orientation of a symmetric complex $\EE$ is an isomorphism $o:\O_{\sX}\cong \det(\EE)$ such that $\det(\theta)=  (-1)^{\frac{r(r-1)}{2}}o \circ o\dual$  where $r=\rank(\EE)$.[This convention is slightly different from [133]. In loc. cit., all symmetric complexes are assumed to be equipped with an orientation.]

* For a morphism $f : \sX \to \sY$ of algebraic stacks, we denote by $\LL_{\sX/\sY}$ the full cotangent complex [85] ([127]) and by $\tau^{\geq -1}\LL_{\sX/\sY}$ the truncated cotangent complex.

* For any algebraic stack $\sX$, we denote by $A_*(\sX)$ the Chow group of Kresch [103] with $\Q$-coefficients. For any $d \in \Q \setminus \Z$, we let $A_d(\cX):=0$.

* For a morphism $f:\sX\to\sY$ and a coherent sheaf (resp. a perfect complex) $E$ on $\sY$, we denote the pullback $f^*E$ (resp. the derived pullback $L f^*E$) by $E|_{\sX}$.

§ STABLE PAIRS

In this section, we introduce the $\PT_q$-stability conditions on 2-dimensional pairs for $q\in\{-1,0,1\}$, and present basic properties which will be used throughout the paper and its sequels. The $\PT_{-1}$-stable pairs coincide with $\DT$-stable pairs (i.e. ideal sheaves), $\PT_1$-stable pairs are $\PT$-stable pairs (i.e. Le Potier stable pairs [135, 136, 128]), and $\PT_0$-stable pairs is a new intermediate notion

\begin{equation*}\label{eq:DTPT0PT1wallcrossing}
\DT:=\PT_{-1} \leadsto \PT_0 \leadsto \PT_{1}=:\PT.
\end{equation*}

Any $\PT_q$ pair $(F,s)$ on a smooth variety $X$ has an associated complex

\[I\udot := [\O_X \xrightarrow{s} F]\]

in $D^b _{\mathrm{coh}}(X)$ — the bounded derived category of coherent sheaves on $X$.
We study $\PT_q$ pairs through these associated complexes. We provide descriptions of these associated complexes under some various assumptions on the singularities of the support of $F$.
The results in this section provide some insights into the geometric meaning of $\PT_q$-stability. They will also play an important role in the sequels to this paper.

Throughout this section, we fix a smooth projective variety $X$ of dimension $n$ over the complex field $\C$.

§.§ $\PT_q$-stability

For any integer $q \geq -1$, consider the two full subcategories

\begin{align*}
\Coh_{\leq q}(X) &:= \{F \in \Coh(X) : \dim(F)\leq q\},\\
\Coh_{\geq q+1}(X) &:=\{ F \in \Coh(X) : T_q(F)=0\}.
\end{align*}

Let $F$ be a coherent sheaf on $X$ with 2-dimensional support and let $s: \O_X \to F$ be a section.
For $q \in \{-1,0,1\}$, we say that the pair $(F,s)$ is $\PT_q$-stable if

\[F \in \Coh_{\geq q+1}(X) \and Q:=\coker(\O_X \xrightarrow{s} F) \in \Coh_{\leq q}(X).\]

By abbreviation, we also refer to $\PT_q$-stable pairs as $\PT_q$ pairs.

The $\PT_q$-stability conditions for the following two extreme cases are well-known.

DT) A pair $(F,s)$ is $\PT_{-1}$-stable if and only if $s:\O_X \to F$ is surjective. Hence $\PT_{-1}$ pairs on $X$ correspond to 2-dimensional closed subschemes of $X$ (which may have irreducible components of dimension 0 and 1). Thus we refer to $\PT_{-1}$-stability as $\DT$-stability.

PT) A pair $(F,s)$ is $\PT_{1}$-stable if and only if $F$ is pure and $\dim(Q)\leq 1$. Hence $\PT_{1}$ pairs are exactly the 2-dimensional stable pairs in the sense of Le Potier <cit.>.[Equivalently, $\PT_{1}$ pairs are the 2-dimensional $\alpha$-stable pairs in the sense of <cit.> for sufficiently big $\alpha(t) \in \Q[t]$.] Their 1-dimensional analogs
are used in [128, 40] (and many other places). Thus we sometimes refer to $\PT_{1}$-stability as $\PT$-stability.

We present some elementary properties of $\PT_q$ pairs. We begin with an observation on the supports.

Given a $\PT_q$ pair $(F,s)$ on $X$, the scheme theoretic support of the coherent sheaf $F$ is the scheme theoretic support of the section $s$.

The proof is analogous to <cit.>.
Let $\mathrm{Ann}(s) \subseteq \O_X$ be the ideal sheaf of the support of $s$. Away from the support of $Q=\coker(s)$, the sheaf $F$ is generated by the section $s$. Hence $\mathrm{Ann}(s)\cdot F \subseteq F$ is set-theoretically supported in the support of $Q$, and thus $\mathrm{Ann}(s)\cdot F \in \Coh_{\leq q}(X)$. Since $F \in \Coh_{\geq q+1}(X)$, we obtain $\mathrm{Ann}(s)\cdot F=0$.

We note that the $\PT_q$ pairs have limit descriptions.

Let $W \subseteq Z \subseteq X$ be closed subschemes such that
$\O_Z \in \Coh_{\geq q+1}(X)$ and $\O_W \in \Coh_{\leq q}(X)$. Suppose $W$ is reduced
and denote its ideal by $\fI \subseteq \O_Z$.
Then $\PT_q$ pairs $(F,s)$ with support $Z$ and cokernel $Q$ satisfying $\Supp(Q)^{\mathrm{red}} \subseteq W$ are equivalent to coherent subsheaves of $\varinjlim \hom_X(\fI^r,\O_Z) / \O_Z$.

The proof is analogous to <cit.>. Choose a $\PT_q$ pair $(F,s)$. By Lemma <ref>, we can form a short exact sequence

\begin{equation}\label{SES.OFQ}
\xymatrix{
0 \ar[r] & \O_Z \ar[r] & F \ar[r] & Q \ar[r] & 0,

where $Z$ is the support of $F$ and $Q$ is the cokernel of $s$. Since $\O_Z \in \Coh_{\geq q+1}(X)$ and $Q \in \Coh_{\leq q}(X)$, we have $\hom_X(Q,\O_Z)=0$. Hence the short exact sequence (<ref>) gives us an inclusion
\[\hom_X(F,\O_Z) \hookrightarrow \hom_X(\O_Z,\O_Z)=\O_Z,\]
which is an isomorphism away from the support of $Q$. Hence $\hom_X(F,\O_Z)$ is an ideal such that $\fI^r \subseteq \hom_X(F,\O_Z)$ for some integer $r$. Since $F \in \Coh_{\geq q+1}(X)$, we have
\[F \subseteq \hom_X(\hom_X(F,\O_Z),\O_Z) \subseteq \hom_X(\fI^r,\O_Z)\subseteq \varinjlim_{r \to \infty}\hom_X(\fI^r,\O_Z),\]
where the last inclusion map follows from $\hom_X(\frac{\fI^r}{\fI^{r+1}},\O_Z)=0$.

On the other hand, choose a coherent subsheaf $F \subseteq \varinjlim \hom_X(\fI^r , \O_Z)$ containing $\O_Z$. Since $F$ is coherent, we have $F \subseteq \hom_X(\fI^r , \O_Z)$ for some sufficiently large $r$. Let $s: \O_X \to \O_Z \hookrightarrow F$ be the composition. Consider a subsheaf $G\subseteq F$ of dimension $\leq q$. By adjunction, the inclusion map $G \hookrightarrow \hom_X(\fI^r,\O_Z)$ corresponds to a map $G \otimes \fI^r \to \O_Z$, which is zero since $\O_Z \in \Coh_{\geq q+1}(X)$. Hence $G=0$. This means that $F \in \Coh_{\geq q+1}(X)$. Therefore $(F,s)$ is a $\PT_q$-stable pair.

Some results in this paper hold for $d$-dimensional pairs. For a coherent sheaf $F$ on $X$ with a $d$-dimensional support, a section $s: \O_X \to F$, and any $q \in \{-1,0,\ldots, d-1\}$, we define the pair $(F,s)$ to be $\PT_q$-stable if

\[F \in \Coh_{\geq q+1}(X) \and Q:=\coker(\O_X \xrightarrow{s} F) \in \Coh_{\leq q}(X).\]

Then Lemma <ref> and Proposition <ref> hold for $d$-dimensional $\PT_q$ pairs as well (as is immediately obvious from the proofs). However, most parts of the paper require $d=2$, so we restrict to this case.

§.§ Special cases

Given a $\PT_0$ pair $(F,s)$ on an $n$-dimensional smooth projective variety $X$, we may consider the following two special cases:

* $(F,s)$ is additionally $\DT$-stable, i.e. $s:\O_X \to F$ is surjective.

* $(F,s)$ is additionally $\PT_1$-stable, i.e. $F$ is pure.

The first case (1) is very clear. It consists of 2-dimensional subschemes $Z \subset X$ with no (possibly embedded) $0$-dimensional components. The second case (2) is more interesting. This case depends on the singularities of the support.

Let $S$ be a pure 2-dimensional subscheme of $X$. Then the following three conditions are equivalent:

$\mathrm{(i)}$ $S$ is not Cohen-Macaulay.

$\mathrm{(ii)}$ $\ext^{n-1}_X(\O_S,K_X) \neq 0$.

$\mathrm{(iii)}$ There exists a $\PT_0$ pair $(F,s)$ with support $S$ and non-zero cokernel $Q$.

Furthermore, for any $m \geq 0$, the $\PT_0$ pairs $(F,s)$ with support $S$ and $\chi(Q) = m$ are in bijective correspondence with the closed points of the Quot scheme

{\curly Quot}_X(\ext^{n-1}_X(\O_S,K_X),m)

parametrizing length $m$ quotients of the 0-dimensional sheaf $\ext^{n-1}_X(\O_S,K_X)$.

Since $S$ is pure, $\ext^i_X(\O_S,K_X)=0$ for $i \neq n -2, n -1$ and $\ext^{n-1}_X(\O_S,K_X)$ is 0-dimensional <cit.>. Let $i: S \hookrightarrow X$ denote the inclusion map. Since $R\hom_X(\O_S,K_X) = R i_* i^! K_X$, we have $\ext^{n-1}_X(\O_S,K_X)=0$ if and only if $S$ is Cohen-Macaulay.

For any 0-dimensional sheaf $Q$, we have

\begin{equation} \label{eqn:ext1iso}
\Ext^1_X(Q,\O_S) \cong \Hom_X(\ext^{n-1}_X(\O_S,K_X), Q^D)
\end{equation}

where $Q^D:=\ext^n_X(Q,K_X)$. Hence (iii) implies (ii). On the other hand, if $\ext^{n-1}_X(\O_S,K_X)\neq 0$, we can always find a closed point $P\in X$ in the support of $\ext^{n-1}_X(\O_S,K_X)$ and a non-zero map $\ext^{n-1}_X(\O_S,K_X) \to \O_P$. By (<ref>), we find a non-split extension in $\Ext^1_X(\O_P^D,\O_S)\neq 0$ from which we obtain a $\PT_0$ pair $(F,s)$ with non-zero cokernel. By Lemma <ref>, the support of this $\PT_0$ pair is $S$. This proves (ii)$\implies$(iii).

Finally, it follows from (<ref>) that there exists a bijection between the collections

\Big( \bigsqcup_{Q} \Ext^1_X(Q,\O_S) \Big) / \cong, \quad \quad \Big( \bigsqcup_Q \Hom_X(\ext^{n-1}_X(\O_S,K_X), Q) \Big) / \cong

where the union is over all 0-dimensional sheaves on $X$ and we divide out by isomorphisms. This bijection is obtained by dualizing the short exact sequence $0 \to \O_S \to F \to Q \to 0$. Since $F$ is pure if and only if $\ext^{n-1}_X(F,K_X)$ is 0-dimensional and $\ext^{n}_X(F,K_X) = 0$ <cit.>, it follows that the short exact sequences with $F$ pure correspond precisely to the surjective morphisms. The result follows.

Now consider a $\PT_1$ pair $(F,s)$. We can similarly consider the following two special cases:

(3) $(F,s)$ is additionally $\PT_0$-stable, i.e. $Q$ is 0-dimensional.

(4) $F$ is additionally reflexive.

Case (3) is exactly case (2) above. Case (4) is a “dual” version of case (1) above, in the sense of the following proposition:

Let $(F,s)$ be a pair of a coherent sheaf $F$ on $X$ and a section $s: \O_X \to F$.

If $(F,s)$ is $\PT_0$-stable and $\chi(F)$ is minimal among all $\PT_0$ pairs with Chern characters $\ch_{n-2}(F)$ and $\ch_{n-1}(F)$, then $(F,s)$ is $\DT$-stable.

If $(F,s)$ is $\PT_1$-stable and $\chi(F)$ is maximal among all $\PT_1$ pairs with Chern characters $\ch_{n-2}(F)$ and $\ch_{n-1}(F)$, then $F$ is reflexive.

In Proposition <ref>, we will show that $\chi(F)$ is bounded below for $\DT$ and $\PT_0$ pairs $(F,s)$ with fixed Chern characters  $\ch_{n-2}(F)$ and $\ch_{n-1}(F)$, and bounded above in the $\PT_1$ case.

(1) Form the short exact sequence

\[\xymatrix{ 0 \ar[r] & \O_Z \ar[r] & F \ar[r] & Q \ar[r] & 0,}\]

where $Z$ is the support of $F$ and $Q$ is the cokernel of $s$. Then we have

\[\ch_{n-2}(\O_Z)=\ch_{n-2}(F), \quad \ch_{n-1}(\O_Z)=\ch_{n-1}(F), \and \chi(\O_Z)=\chi(F)-\chi(Q).\]

Since $(\O_Z,1:\O_X \twoheadrightarrow \O_Z$) is a $\PT_0$ pair, and
$\chi(F)$ is minimal, we have $Q=0$. Therefore $(F,s)$ is a $\DT$ pair.

(2) Consider the canonical map

\[\theta_F : F \to F^{DD}.\]

The purity of $F$ implies $\ext^{n-1}_X(F,K_X) \in \Coh_{\leq 0}(X)$ and $\ext^n_X(F,K_X)=0$ by <cit.>. Moreover, the spectral sequence

\[\ext^p_X(\ext^{-q} _X(F,K_X),K_X) \Rightarrow F \]

in <cit.> gives us a short exact sequence

\[\xymatrix{0 \ar[r] & F \ar[r] & F^{DD} \ar[r] & \ext_X^{n-1}(F,K_X)^D \ar[r] & 0,}\]

where $\ext_X^{n-1}(F,K_X)^D \in \Coh_{\leq 0}(X)$. Hence $(F^{DD}, \theta_F \circ s)$ is also a $\PT_1$ pair such that

\[\chi(F^{DD}) = \chi(F) +\chi(\ext^{n-1}_X(F,K_X)^D) \geq \chi(F).\]

Since $\chi(F)$ is maximal, we have $F=F^{DD}$.

§.§ Associated complexes

Let $X$ be a smooth projective variety of dimension $n$. Given a $\PT_q$ pair $(F,s)$ on $X$, we can form an associated complex

\[I\udot:=[\O_X \xrightarrow{s} F],\]

with $\O_X$ in degree zero. We view $I\udot$ as an element of $D^b_{\mathrm{coh}}(X)$ — the bounded derived category of coherent sheaves on $X$. In this subsection, we analyze $\PT_q$ pairs in terms of the associated complex $I\udot$.

We first note that a $\PT_q$ pair is equivalent to its associated complex.

Assume $X$ is a smooth projective variety of dimension $n \geq 4$.
Consider two $\PT_q$ pairs $(F_1,s_1)$ and $(F_2,s_2)$. Then the two associated complexes $I_1\udot:=[\O_X \xrightarrow{s_1} F_1]$ and $I_2\udot:=[\O_X \xrightarrow{s_2} F_2]$ are quasi-isomorphic if and only if the two $\PT_q$ pairs $(F_1,s_1)$ and $(F_2,s_2)$ are isomorphic.

The proof is analogous to <cit.>. Since

\[\Ext^{\leq 1}_X(F_1,\O_X)\cong H^{\geq n-1}(X,F_1\otimes K_X)^*=0,\]

by $F_1 \in \Coh_{\leq2}(X)$, the map

\[\C=\Hom_X(\O_X,\O_X) \to \Hom_X(I^\mdot_1,\O_X)\]

is an isomorphism. Therefore, any quasi-isomorphism $\phi:I^\mdot_1 \to I^\mdot_2$ can be completed to an isomorphism of distinguished triangles

\[\xymatrix{
I^\mdot_1 \ar[r] \ar[d]^{\phi} & \O_X \ar[r]^{s_1}<EMAIL_ADDRESS>& F_1 \ar[r]<EMAIL_ADDRESS>&  \\
I^\mdot_2 \ar[r]  & \O_X \ar[r]^{s_2} & F_2 \ar[r] &

for some dotted arrows $c$ and $\psi$. Since $c\in \Gamma(X,\O_X^*)=\C^*$ is a non-zero constant, $c^{-1}\cdot\psi :F_1 \to F_2$ is an isomorphism of sheaves with $\psi(s_1)=s_2$.

We show that a $\PT_0$ pair, for which the pure part of its scheme theoretic support is Cohen-Macaulay, can be expressed by a map from a $\DT$ pair to a $1$-dimensional pure sheaf.

For any $\PT_0$ pair $(F,s)$ on $X$ such that $S=\Supp(F)^{\pure}$ is Cohen-Macaulay, we have a natural quasi-isomorphism

\begin{equation}\label{Eq.PT0.ItoG}
I\udot \cong [I_{S/X} \to T_1(F)],
\end{equation}

where $T_1(F)$ is zero or pure 1-dimensional and $I_{S/X}$ denotes the ideal sheaf of $S$ in $X$.

In particular, when $\dim(X) = 3$, we have $I_{S/X} = \O_X(-S)$ and

\[I\udot \cong [\O_X \to T_1(F)(S)](-S)\]

which, for $T_1(F) \neq 0$, is a 1-dimensional stable pair on $X$ as in [128].

We may assume $T_1(F) \neq 0$, because otherwise the result is immediate from Proposition <ref>. Note that $F/T_1(F)$ is a pure 2-dimensional sheaf and $T_1(F)$ is a pure 1-dimensional sheaf. Hence the support of the $\PT_0$-stable pair $\O_X \to F/T_1(F)$ is also pure. Let $Z = \Supp(F)$, so $S = Z^{\pure}$. Since $T_1(\O_Z)=I_{S/Z}$ by definition,
the support of the $\PT_0$ pair $\O_X \to F/T_1(F)$ is $S$. By Proposition <ref> above, we obtain the desired isomorphism $\O_S \cong F/T_1(F)$. Moreover, since the commutative square

\[\xymatrix{
I_{S/X} \ar[r] \ar[d] & T_1(F) \ar[d] \\
\O_X \ar[r] & F

is cartesian and cocartesian, we have the desired quasi-isomorphism (<ref>).

Dually, $\PT_1$ pairs (with Cohen-Macaulay support) can be expressed by maps between $1$-dimensional pure sheaves and $\DT$ pairs.

For any $\PT_1$ pair $(F,s)$ on $X$ with Cohen-Macaulay support $S$, we have a natural quasi-isomorphism

\[I\udot \cong \cone(Q[-2] \to I_{S/X}), \]

where the cokernel $Q$ is zero or pure 1-dimensional.

The quasi-isomorphism follows from the canonical distinguished triangle

\[\xymatrix{ I_{S/X} \ar[r] & I^\mdot \ar[r] & Q[-1] \ar[r] & I_{S/X}[1]. }\]

Thus it suffices to assume $Q \neq 0$ and show it is pure 1-dimensional.
Applying $\Hom_X(T_0(Q),-)$ to the short exact sequence $0 \to \O_S \to F \to Q \to 0$, we obtain a long exact sequence

\[\xymatrix{ \cdots \ar[r] & \Hom_X(T_0(Q),F) \ar[r] & \Hom_X(T_0(Q),Q) \ar[r] & \Ext^1_X(T_0(Q), \O_S) \ar[r] & \cdots }.\]

By purity of $F$ and Proposition <ref>, we deduce $T_0(Q)=0$, which implies that $Q$ is pure 1-dimensional.

For $\PT_1$ pairs with Gorenstein support, we also have the following characterization.

For any $\PT_1$ pair $(F,s)$ on $X$ with Gorenstein support $S$ and non-zero cokernel $Q$, we have a natural quasi-isomorphism

\begin{equation}\label{Eq.PT1toPTc}
F\dual \otimes K_X[n-2] \cong [\O_S \to Q^D \otimes K_S\dual] \otimes K_S
\end{equation}

where $Q^D:=\ext^{n-1}_X(Q,K_X)$ is a pure 1-dimensional sheaf supported on $S$.
Moreover, this induces an equivalence between $\PT_1$ pairs supported on $S$ with non-zero cokernel and 1-dimensional $\PT$ pairs on $S$.

Since $S$ is Gorenstein, we have $R\hom_X(\O_S,K_X) \cong K_S[-(n-2)]$.
Moreover, $Q$ is pure 1-dimensional by Proposition <ref>.
Applying the functor $R\hom_X(-,K_X)$ to the short exact sequence $0 \to \O_S \to F \to Q \to 0$, we obtain a distinguished triangle

\[\xymatrix{ F\dual \otimes K_X[n-2] \ar[r] & K_S \ar[r] & Q^D \ar[r] & F\dual \otimes K_X[n-1], }\]

which implies the desired quasi-isomorphism (<ref>).

Since $Q$ is pure 1-dimensional, we have $Q \cong Q^{DD}$ and $Q^D$ is pure 1-dimensional as well. Moreover $F$ is pure, hence the cokernel $\ext_X^{n-1}(F,K_X)$ of the map $K_S \to Q^D$ is 0-dimensional. It is easy to see that all these operations are invertible thereby establishing an equivalence.

The equivalence in Proposition <ref> implies that $\PT_1$ pairs on Gorenstein (and smooth) supports have simple descriptions. Fix a 2-dimensional pure subscheme $S$ of $X$.

Assume that $S$ is Gorenstein. A $\PT_1$ pair $(F,s)$ on $X$ with support $S$ and non-zero cokernel is equivalent to a pair $(C,t)$ where $C \subset S$ is a pure 1-dimensional subscheme and $t : \omega_C \twoheadrightarrow R$ is a 0-dimensional quotient.

By the previous lemma, we may view a $\PT_1$ pair on $X$ with support $S$ and non-zero cokernel as a 1-dimensional $\PT$ pair on $S$. The support of this stable pair is a 1-dimensional pure subscheme $C \subset S$ and the stable pair is determined by a short exact sequence

0 \to \O_C \to G \to Q \to 0,

where $G$ is pure 1-dimensional and $Q$ is 0-dimensional. Applying $R \hom_S(-, K_S)$ and using $\ext^2_S(G,K_S) = 0$ (purity of $G$), we obtain a quotient

t : \omega_C = \ext_S^1(\O_C,K_S) \twoheadrightarrow \ext^2_S(Q,K_S).

Note that applying $R \hom_S(-, K_S)$ again, and using the fact that $\O_C^{DD} \cong \O_C$ and $\O_C^D$ is pure 1-dimensional, we can recover the original 1-dimensional $\PT$ pair on $S$, and hence $\PT_1$ pair on $X$.

Assume that $S$ is smooth. A $\PT_1$ pair $(F,s)$ on $X$ with support $S$ and non-zero cokernel is equivalent to a pair $(C,Z)$ of an (non-zero) effective divisor $C \subseteq S$ and a 0-dimensional subscheme $Z \subseteq C$.

Since any pure 1-dimensional subscheme $C$ of a smooth surface $S$ is a local complete intersection, $C$ is Gorenstein and $K_C$ is invertible. Hence the proof follows directly from Corollary <ref>.

§.§ Numerical bounds

Let $X$ be a smooth projective variety of dimension $n$ with fixed very ample line bundle $\O_X(1)$. For any coherent sheaf $F$ on $X$, we denote its Hilbert polynomial by

\[P_F(t) = \sum_{i\geq0} \alpha_i(F) \cdot \frac{t^i}{i!},\]

where $\alpha_0(F)=\chi(F)$ is the Euler characteristic. In this subsection, we derive boundedness results on $\alpha_i(F)$ for $\PT_q$ pairs $(F,s)$ which are crucial in the GIT construction in subsection <ref>.

Fix rational numbers $\alpha_2>0$ and $\alpha_1$.

$(1)$ For each $q\in \{-1,0,1\}$, the set

\[\{\alpha_1(F) : \text{$\PT_q$ pairs $(F,s)$ such that $\alpha_2(F)=\alpha_2$}\}\]

is bounded below.

$(2)$ The two sets

\[\{\chi(F) : \text{$\DT$ pairs $(F,s)$ such that $\alpha_2(F)=\alpha_2$, $\alpha_1(F)=\alpha_1$}\}\]

\[\{\chi(F) : \text{$\PT_0$ pairs $(F,s)$ such that $\alpha_2(F)=\alpha_2$, $\alpha_1(F)=\alpha_1$}\}\]

are bounded below, and the set

\[\{\chi(F) : \text{$\PT_1$ pairs $(F,s)$ such that $\alpha_2(F)=\alpha_2$, $\alpha_1(F)=\alpha_1$}\}\]

is bounded above.

(1) The $\DT$ case is shown in <cit.>. The $\PT_0$ case follows directly from the $\DT$ case by considering the short exact sequence $0 \to \O_Z \to F \to Q \to 0$ since $\alpha_2(F)=\alpha_2(\O_Z)$ and $\alpha_1(F)=\alpha_1(\O_Z)$. Finally, the $\PT_1$ case also follows directly from the $\DT$ case by considering the short exact sequence $0 \to \O_S \to F \to Q \to 0$ since $\alpha_2(F)=\alpha_2(\O_S)$ and $\alpha_1(F)\geq \alpha_1(\O_S)$.

(2) ($\DT$) Choose a $\DT$ pair $(\O_Z,1:\O_X \twoheadrightarrow \O_Z)$ such that $\alpha_2(\O_Z) = \alpha_2$, $\alpha_1(\O_Z) = \alpha_1$, and $\chi(\O_Z)<0$. Consider the map

\begin{equation}\label{2}
(X\setminus Z)^{[-\chi(\O_Z)]} \to \Hilb(X,\tfrac12\alpha_2 t^2+ \alpha_1 t) : W \mapsto Z \sqcup W
\end{equation}

from the Hilbert scheme of points to the Hilbert scheme of closed subschemes with Hilbert polynomial $v=\tfrac12\alpha_2 t^2+ \alpha_1 t$. Since the map (<ref>) is injective and the Hilbert scheme $\Hilb(X,v)$ is of finite type, the Euler characteristic $\chi(\O_Z)$ has a lower bound which only depends on $X$, $\alpha_2$, and $\alpha_1$.

($\PT_0$) Given a $\PT_0$ pair $(F,s)$, we can form a short exact sequence

\[\xymatrix{0 \ar[r] & \O_Z \ar[r] & F \ar[r] & Q \ar[r] & 0,}\]

where $Z$ is the support of $F$ and $Q$ is the cokernel of $s$. Since

\[\alpha_2(F)=\alpha_2(\O_Z), \quad \alpha_1(F)=\alpha_1(\O_Z), \and \chi(F) =\chi(\O_Z)+\chi(Q),\]

a lower bound in the case ($\DT$) also gives us a lower bound for the case ($\PT_0$).

($\PT_1$). Choose a $\PT_1$ pair $(F,s)$ and form a short exact sequence

\[\xymatrix{0 \ar[r] & \O_S \ar[r] & F \ar[r] & Q \ar[r] & 0,}\]

where $S$ is the support of $F$ and $Q$ is the cokernel of $s$. By Lemma <ref>(2) below, it suffices to show that $\mu_{\min}(F)$ is bounded below. Note that there exists a surjective map $F \twoheadrightarrow F_{\min}$ to a 2-dimensional $\mu$-semistable sheaf $F_{\min}$ such that $\mu_{\min}(F)=\mu(F_{\min})$. We can assume that $F_{\min}$ is pure. We can form an induced short exact sequence

\[\xymatrix{0 \ar[r] & \O_Z \ar[r] & F_{\min} \ar[r] & R \ar[r] & 0}\]

for a subscheme $Z \subseteq S$ and a quotient sheaf $Q \twoheadrightarrow R$. Note that we have

\[\mu_{\min}(\O_S) \leq \mu_{\min}(\O_Z) \leq \mu(F_{\min}) = \mu_{\min}(F)\]

(cf. <cit.>). Since $S$ is pure, the minimal slope $\mu_{\min}(\O_S)$ is bounded below by <cit.>. Therefore $\mu_{\min}(F)$ is also bounded below.

We need the following lemma to complete the proof of Proposition <ref>(2) for $\PT_1$ pairs.

Let $X$ be a smooth projective variety with a fixed very ample line bundle $\O_X(1)$ and let $\alpha_2$, $\alpha_1$ be rational numbers such that $\alpha_2>0$.

$(1)$ The set of integers

\[\left\{
\alpha_0 :
\begin{matrix}
\text{$\mu$-stable reflexive sheaves $F$ on $X$ such that} \\
\text{$P_F(t) = \tfrac{1}{2} \alpha_2 t^2 +\alpha_1 t +\alpha_0$}
\end{matrix}\right\}\]

is bounded above.

$(2)$ For a fixed rational number $C$, the set of integers

\[\left\{
\alpha_0 :
\begin{matrix}
\text{pure sheaves $F$ on $X$ such that} \\
\text{$P_F(t) = \tfrac{1}{2} \alpha_2 t^2 +\alpha_1 t +\alpha_0$ and $\mu_{\min}(F)\geq C $}
\end{matrix}\right\}\]

is bounded above.

(1) The proof is similar to <cit.>. Choose a $\mu$-stable reflexive sheaf $F$ with $P_F(t) = \tfrac{1}{2} \alpha_2 t^2 +\alpha_1 t +\alpha_0$. Assume that $\alpha_0>0$. Let $Z$ be the support of $F$. Let $U \subseteq Z^{\mathrm{red}}$ be a non-empty smooth connected open subscheme of the reduced subscheme $Z^{\mathrm{red}}$ of $Z$ . After shrinking $U$ to a possibly smaller non-empty open subscheme, we may assume that $F|_U \cong \O_U ^{\oplus r}$ is a trivial vector bundle on $U$.

We first claim that there is a map

\[\Phi : U^{[\alpha_0]} \to M^{\mu}(X,\tfrac{1}{2} \alpha_2 t^2 +\alpha_1 t)\]

from the Hilbert scheme of points on $U$ to the moduli space of $\mu$-stable pure sheaves on $X$ such that there is a short exact sequence

\begin{equation}\label{eq:2}
\xymatrix{0 \ar[r] & \Phi(W) \ar[r] & F \ar[r] & \O_W \ar[r] & 0,}
\end{equation}

for all $W \in U^{[\alpha_0]}$. Indeed, let $\cW \subseteq U^{[\alpha_0]}\times U \subseteq U^{[\alpha_0]}\times X$ be the universal family. Then we have a surjective map $\O_{U^{[\alpha_0]}} \boxtimes F|_U \twoheadrightarrow \O_{U^{[\alpha_0]} \times U} \twoheadrightarrow \O_{\cW}$ induced by some projection map $\O_U^{\oplus r} \to \O_U$. Hence we can form a short exact sequence

\begin{equation}\label{eq:3}
\xymatrix{0 \ar[r] & \F' \ar[r] & \O_{U^{[\alpha_0]}} \boxtimes F \ar[r] & \O_{\cW} \ar[r] & 0}
\end{equation}

of sheaves on $U^{[\alpha_0]}\times X$, which are flat over $U^{[\alpha_0]}$. Hence $\F'$ defines the map $\Phi$ and by taking the fibres of (<ref>) over $W \in U^{[\alpha_0]}$, we obtain (<ref>).
From (<ref>) and the fact that $F$ is reflexive, it follows that $\Phi(W)$ is pure 2-dimensional and $\Phi(W)^D \cong F^D$.
Moreover, $\Phi(W) \cong F \in \Coh_{2,1}(X):=\Coh_{\leq2}(X)/\Coh_{\leq0}(X)$ so that $\Phi(W)$ is $\mu$-stable.

We then claim that the map $\Phi$ is injective. Choose two 0-dimensional closed subschemes $W_1, W_2 \subseteq U$ and let $F_i=\Phi(W_i)$ for both $i=1,2$. Assume that there is an isomorphism $\phi : F_1 \xrightarrow{\cong} F_2$ of sheaves. Then we can form a commutative square

\[\xymatrix{ F_1 \ar[d]^{\phi} \ar@{^{(}->}[r] & F_1^{DD} \ar[d]^{\phi^{DD}} & F \ar[l]_-{\cong}<EMAIL_ADDRESS>F_2 \ar@{^{(}->}[r] & F_2^{DD} & F \ar[l]_-{\cong} }\]

since $F_1^{DD} \cong  F^{DD} \cong F_2^{DD}$ and $F$ is reflexive. Since $F$ is $\mu$-stable, $F$ is stable, and thus simple, i.e. $\Hom_X(F,F)=\C$. Therefore by replacing $\phi$ by $c\cdot\phi$ for some constant $c \in \C^*$, we may assume that the dotted arrow $\psi$ is the identity map $\id_F$.
Hence $F_1 \cong F_2$ as subsheaves of $F$, and thus the cokernels are isomorphic, $\O_{W_1} \cong \O_{W_2}$. Therefore $W_1=W_2$ and $\Phi$ is injective.

Since $\Phi$ is injective and $\dim(U^{[\alpha_0]}) \geq \dim(U) \cdot\alpha_0$, the Euler characteristic $\alpha_0$ is bounded above by the dimension of the finite type scheme $M^{\mu}(X,\tfrac12 \alpha_2 t^2 + \alpha_1 t)$. This proves (1).

(2) Case 1. We first show that

\[
\{\alpha_0: \text{ $\mu$-stable pure sheaves $F$ on $X$ such that $P_F(t) = \tfrac{1}{2} \alpha_2 t^2 +\alpha_1 t +\alpha_0$}\}
\]

is bounded above. Let $F$ be a $\mu$-stable pure sheaf with $P_F(t) = \tfrac{1}{2} \alpha_2 t^2 +\alpha_1 t +\alpha_0$. Since $F$ is pure, we can form a short exact sequence

\[\xymatrix{0 \ar[r] & F \ar[r] & F^{DD} \ar[r] & R \ar[r] & 0 }\]

for some 0-dimensional sheaf $R$ by <cit.>. Then $F^{DD}$ is a reflexive $\mu$-stable sheaf with $P_{F^{DD}}(t) = \tfrac{1}{2} \alpha_2 t^2 +\alpha_1 t + (\alpha_0+\chi(R))$. Since $\alpha_0(F^{DD})= \alpha_0 + \chi(R)$ is bounded above by Lemma <ref>(1), $\alpha_0(F)$ is also bounded above.

Case 2. We then show that

\[
\{\alpha_0 : \text{ $\mu$-semistable pure sheaves $F$ on $X$ such that $P_F(t) = \tfrac{1}{2} \alpha_2 t^2 +\alpha_1 t +\alpha_0$}\}
\]

is bounded above. Choose a $\mu$-semistable pure sheaf $F$ with $P_F(t) = \tfrac{1}{2} \alpha_2 t^2 +\alpha_1 t +\alpha_0$. Consider the Jordan-Hölder filtration

\[0=F_0 \subseteq F_1 \subseteq \cdots \subseteq F_l =F\]

of $F$ with respect to $\mu$-stability <cit.>. Since $F$ is pure, we can assume that all $F_i/F_{i-1}$ are also pure. Since each $F_i/F_{i-1}$ is a $\mu$-stable pure sheaf with the slope $\mu(F_i/F_{i-1})=\mu(F)$, the result in Case 1 proves Case 2 by induction.

Case 3. Finally, we consider the general case. Let $F$ be a pure sheaf with $P_F(t)= \tfrac12 \alpha_2 t^2 + \alpha_1 t + \alpha_0$. Choose a Harder-Narasimhan filtration

\[0=F_0 \subseteq F_1 \subseteq \cdots \subseteq F_l =F\]

of $F$ with respect to the $\mu$-stability <cit.>. Since $F$ is pure, we may assume that $F_i/F_{i-1}$ are $\mu$-semistable pure sheaves. If $\alpha_2(F)$, $\alpha_1(F)$ are fixed and $\mu_{\min}(F)$ is bounded below, then there are only finitely many possible $\alpha_1(F_i/F_{i-1})$ (and $\alpha_2(F_i/F_{i-1})$). Therefore

\[\alpha_0(F) = \sum_i \alpha_0(F_i/F_{i-1})\]

is also bounded above by the result in Case 2.

§ MODULI SPACES

In this section, we construct moduli spaces of $\PT_q$ pairs via geometric invariant theory and prove that they are open subschemes of the moduli space of perfect complexes parametrizing polynomially Bridgeland stable objects.

§.§ Moduli stack of pairs

In this subsection, we introduce the moduli stack of all pairs and prove that the $\PT_q$-stability conditions are bounded and open conditions.

Let $X$ be a smooth projective variety over $\C$.

* We define the moduli stack of pairs on $X$ as a 2-functor

\[\tPair(X) : \mathrm{Sch}^{\mathrm{op}}_{\C} \to \mathrm{Groupoid}\]

sending a scheme $T$ to the groupoid of (not necessarily 2-dimensional) pairs $(F,s)$ of $T$-flat coherent sheaves $F$ on $X \times T$ and sections $s \in \Gamma(X\times T, F)$. The morphisms $(F_1,s_1) \to (F_2,s_2)$ in the groupoid are isomorphisms $\phi :F_1 \to F_2$ of sheaves such that $\phi(s_1)=s_2$.

* We define the moduli stack of non-zero pairs on $X$ as the substack

\[\cPair(X) \subseteq \tPair(X)\]

consisting of the pairs $(F,s)$ such that $s_t \neq 0 \in \Gamma(X,F_t)$ for all $t \in T$.

The moduli stacks $\tPair(X)$ and $\cPair(X)$ are algebraic stacks. Moreover, they have explicit descriptions as unions of open substacks which are global quotient stacks. We will now review the proofs of these facts.

Let us first fix some notation. Let $X$ be a smooth projective variety of dimension $n$ with a fixed very ample line bundle $\O_X(1)$. Fix $v \in H^*(X,\Q)$ and let $P_v(t) \in \Q[t]$ be the associated Hilbert polynomial. Choose an integer $m$. Fix a vector space $V$ of dimension $P_v(m)$. Let

\begin{equation}\label{eq:def.quot}
\cQ :=\cQuot(X,V \otimes \O_X(-m),v)
\end{equation}

be the Quot scheme of quotients $\rho : V\otimes \O_X(-m) \twoheadrightarrow F$ with $\ch(F)=v$.

\begin{equation}\label{eq:quot.universal}
V \otimes \O_{\cQ} \otimes \O_X(-m) \twoheadrightarrow \FF
\end{equation}

be the universal quotient on $\cQ\times X$ and let $\pi : \cQ \times X \to \cQ$ be the projection map. Consider the open subscheme

\begin{equation}\label{eq:def.quot0}
\cQ^{\circ} := \Big\{(F,\rho) \in \cQ : V \stackrel{\cong}{\to} H^0(X,F(m)) \text{ and } H^{i}(X,F(m))=0 \, \forall i>0\Big\},\
\end{equation}

where $V \to H^0(X,F(m))$ is the map induced by $\rho$. Then the canonical map

\begin{equation}\label{eq:S2.4}
V \otimes \O_{\cQ} \to R\pi_*(\FF(m))
\end{equation}

induced by (<ref>) is an isomorphism over $\cQ^{\circ}$.

The moduli stack $\cPair(X)$ has the following properties:

$(1)$ We have a decomposition

\[\cPair(X) = \bigsqcup_{v \in H^*(X,\Q)} \cPair(X,v)\]

by disjoint open substacks $\cPair(X,v)$ consisting of the pairs $(F,s)$ on $X$ with $\ch(F)=v \in H^*(X,\Q)$.

$(2)$ We have

\[\cPair(X,v) = \bigcup_{m \geq 0}\cPair(X,v)_m\]

for open substacks $\cPair(X,v)_m$ consisting of the pairs $(F,s)$ on $X$ such that $F(m)$ is globally generated and $H^{>0}(X,F(m))=0$.

$(3)$ We have a global quotient stack presentation

\[\cPair(X,v)_m = \left[\PP_{\cQ^{\circ}}(R \pi_* \FF)/\PGL(V)\right].\]

where $\PP_{\cQ^{\circ}}(R\pi_*\FF):=\Proj \Sym h^0((R \pi_* \FF)\dual)$ denotes the projective cone.

Consequently, the moduli stack $\cPair(X)$ is an algebraic stack (locally of finite type).

Let $\cCoh(X)$ denote the moduli stack of coherent sheaves on $X$. Then

\[\cCoh(X) = \bigsqcup_{v \in H^*(X,\Q)} \cCoh(X,v) \and \cCoh(X,v) = \bigcup_{m \geq 0} \cCoh(X,v)_m\]

for open substacks $\cCoh(X,v)_m \subseteq \cCoh(X,v) \subseteq \cCoh(X)$
such that

\begin{align*}
\cCoh(X,v)(\C) &= \{F \in \cCoh(X)(\C) : \ch(F)=v \in H^*(X,\Q) \},\\
\cCoh(X,v)_m(\C) &= \left\{F \in \cCoh(X,v)(\C) : \begin{matrix} F(m) \text{ globally generated and } \\  H^i(X,F(m))=0 \text{ for all } i>0 \end{matrix} \right\}.
\end{align*}

Let $\FF^C$ be the universal family on $\cCoh(X) \times X$ and let $\pi^C : \cCoh(X) \times X \to \cCoh(X)$ be the projection map. Then $R\pi^C_*(\FF^C(m))$ is a vector bundle over $\cCoh(X,v)_m$ and the associated $\GL_{P_v(m)}$ bundle is exactly $\cQ^{\circ}$. Hence we have a global quotient stack presentation (see also <cit.>)

\[\cCoh(X,v)_m = \left[\cQ^{\circ} / \GL(V) \right].\]

By definition, the moduli stack of pairs is the abelian cone

\[\tPair(X) = C(\ext^n_{\pi^C}(\FF^C,K_X)) = \Spec \Sym  \ext^n_{\pi^C}(\FF^C,K_X) \]

associated to the coherent sheaf $\ext^n_{\pi^C}(\FF^C,K_X)=h^0((R\pi^C_*\FF^C)\dual)$ on $\cCoh(X)$ and the moduli stack of non-zero pairs is the complement

\[\cPair(X) = \tPair(X) \setminus 0_{\cCoh(X)} \subseteq \tPair(X)\]

of the zero section $0_{\cCoh(X)}: \cCoh(X) \hookrightarrow \tPair(X)$. Hence, we have  (1) and (2) for the open substacks $\cPair(X,v):=\cPair(X)\times_{\cCoh(X)}\cCoh(X,v)$ and $\cPair(X,v)_m:=\cPair(X)\times_{\cCoh(X)}\cCoh(X,v)_m$. Moreover, we have a global quotient stack presentation

\[\cPair(X,v)_m = \left[(C_{\cQ^{\circ}}(\ext^n_\pi(\FF,K_X))\setminus 0_{\cQ^{\circ}})/\GL(V) \right] = \left[\PP_{\cQ^{\circ}}(R\pi_*\FF)/\PGL(V)\right]\]

since $\PGL(V) = \GL(V)/\GG_m$.

There exists an open substack $\cPair^{(q)}(X,v)$ of $\cPair(X,v)$, for any $q \in \{-1,0,1\}$, such that the $\C$-points are given by

\[\cPair^{(q)}(X,v)(\C) = \{(F,s)\in \cPair(X,v) : \text{$(F,s)$ is $\PT_q$-stable}\}\]

Moreover, the moduli stack $\cPair^{(q)}(X,v)$ is an algebraic space of finite type.

We will prove Proposition <ref> through  three steps. We first prove that $\PT_q$ pairs are simple pairs.

For any $\PT_q$ pair $(F,s)$ on $X$, we have

Consider the short exact sequence of sheaves

\[\xymatrix{0 \ar[r] & \O_Z \ar[r] & F \ar[r] & Q \ar[r] & 0}\]

where $Z$ is the support of $F$ and $Q$ is the cokernel of $s$. By the $\PT_q$-stability condition, we have $\Hom_X(Q,F) =0.$ Hence we have an injective map

\begin{equation}\label{eq:S2.15}
\Hom_X(F,F) \longrightarrow \Hom_X(\O_Z,F) = \Gamma(X,F) : \phi \mapsto \phi(s).
\end{equation}

Since $\Aut_{\cPair(X)}(F,s)$ is the fibre of the above map (<ref>) over the point $s \in \Gamma(X,F)$, the automorphism group $\Aut_{\cPair(X)}(F,s)$ is trivial.

Secondly, we prove that the collection of $\PT_q$ pairs is bounded.

The collection of sheaves

\[\{F \in \cCoh(X,v) (\C): \text{$\exists s\in \Gamma(X,F)$ such that $(F,s)$ is $\PT_q$-stable}\}\]

for fixed $X$ and $v$ is bounded in the sense of <cit.>.

The boundedness of the $\DT=\PT_{-1 }$ pairs follows from <cit.> and the boundedness of the $\PT=\PT_{1}$ pairs follows from <cit.>. Thus it suffices to consider the $\PT_0$ pairs.

Given a $\PT_0$ pair $(F,s)$ with $\ch(F) = v$, let $Z$ be the support of $F$ and $Q$ the cokernel of $s$. By Proposition <ref>(2) for $\DT$-stability, $\chi(\O_Z)$ is bounded below. On the other hand $\chi(\O_Z)$ is bounded above since $\chi(Q)\geq 0$. Moreover, $\ch_i(\O_Z)=\ch_i(F)$ for $i<n$. Hence there are only finitely many possible Chern characters $\ch(\O_Z)$, and the collection of all such $\O_Z$ is bounded by <cit.>.
Therefore, there exists an integer $m$ such that for all $\O_Z$ as above, we have $H^i(X,\O_Z(m-i)) = 0$ for all $i>0$. For any $\PT_0$ pair $(F,s)$ with $\ch(F) = v$, we then also have $H^i(X,F(m-i)) = 0$ for all $i>0$ ($H^{i}(X,Q(m-i))$ is automatically zero since $Q$ is 0-dimensional). By <cit.>, we deduce the required boundedness for $\PT_0$ pairs.

The proof of Lemma <ref> also proves the following lemma, which will be used in the proof of Proposition <ref> in the next subsection.

The collection of sheaves

\[\{F \in \cCoh(X,v)(\C) : \text{$\exists s\in \Gamma(X,F)$ such that $\coker(\O_X \xrightarrow{s} F)\in \Coh_{\leq0}(X)$}\}\]

for fixed $X$ and $v$ is bounded.

Lastly, we show that the $\PT_q$-stability conditions are open.

Let $(F,s)$ be a $T$-flat family of pairs on $X$ parametrized by a scheme $T$ of finite type over $\C$. Then

\[\{t \in T(\C) : \text{the fibre $(F_t,s_t)$ of $(F,s)$ is $\PT_q$-stable}\}\]

is a Zariski open subset of $T(\C)$.

We first show that the condition

\[Q_t:=\coker(\O_X \xrightarrow{s_t} F_t) \in \Coh_{\leq q}(X)\]

is an open condition. By <cit.>, the set $\{P_{Q_t} : t \in T(\C)\}$ of Hilbert polynomials is finite. Hence the result follows from upper semi-continuity of Hilbert polynomials of non-flat families of coherent sheaves.

Now we consider the second condition

\[F_t \in \Coh_{\geq q+1}(X).\]

Firstly, if $q=-1$, then this condition is always satisfied. Secondly, if $q=1$, this condition is open by the openness of purity <cit.>. Finally, assume that $q=0$. We show that the set of the Hilbert polynomials

\begin{equation} \label{eqn:PF/T0F}
\{P_{{F_t}/{T_0(F_t)}} : t \in T(\C)\}
\end{equation}

is finite. It suffices to show that $\chi(T_0(F_t))$ is bounded above. Note that the short exact sequence $0 \to T_0 (F_t)\to F_t \to F_t/T_0(F_t) \to 0$
induces a short exact sequence

\[\xymatrix{0 \ar[r] & \ext^n_X(F_t/T_0(F_t), K_X) \ar[r] & \ext^n_X(F_t,K_X) \ar[r] & \ext^n_X(T_0(F_t),K_X) \ar[r] & 0}\]

of 0-dimensional sheaves by <cit.>. Then we have

\[\chi(T_0(F_t)) = \chi(\ext^n_X(T_0(F_t),K_X)) \leq \chi(\ext^n_X(F_t,K_X)) \leq \dim_\C \Ext^n_X(F_t,K_X)\]

where $\pi^T : X \times T \to T$ is the projection map, $i_t :\{t\} \hookrightarrow T$ is the inclusion map, and the last inequality follows from the local-to-global spectral sequence. Since $\Ext^n_X(F_t,K_X)$ are the fibres of the coherent sheaf  $\ext^n_{\pi^T} (F,K_X)$ on a scheme $T$ of finite type, their dimensions are bounded above.

Since, for any $t \in T(\C)$, the relative Quot schemes

\[\cQuot(X\times T/T , F, P_{F_t/T_0(F_t)})  \to T\]

are proper over $T$, their images are closed. The claim follows from finiteness of (<ref>).

Proposition <ref> follows directly from the above results.

By Lemma <ref>, we can form $\cPair^{(q)}(X,v)$ as an open substack of $\cPair(X,v)$. Then Lemma <ref> proves that $\cPair^{(q)}(X,v)$ is an algebraic space and Lemma <ref> proves that $\cPair^{(q)}(X,v)$  is of finite type.

By [70, 135], the moduli spaces $\cPair^{(q)}(X,v)$ of $\PT_q$ pairs are projective schemes for $q=-1$ and $q=1$. In the next subsection, we will show that the space $\cPair^{(0)}(X,v)$ is also a projective scheme.

We end this subsection with a remark on the natural derived enhancement of the moduli stack of pairs. This derived structure will not be used in the rest of the paper.

There is a derived stack  $R\tPair(X)$ whose classical truncation is $\tPair(X)$ and the tangent complex at a point $(F,s)$ is

\[\TT_{R\tPair(X)}|_{(F,s)} = R\Hom_X(I^\mdot,F)\]

where $I^\mdot=[\O_X \xrightarrow{s} F]$. We sketch the construction here. Recall that there exists a derived moduli stack $R\cCoh(X)$ of coherent sheaves as an open substack of the derived moduli stack $R\cPerf(X)$ of perfect complexes. Let $\FF$ denote the universal family on $R\cCoh(X)$ and let $\pi:R\cCoh(X)\times X\to R\cCoh(X)$ denote the projection map. Define the derived moduli stack of pairs as the total space

\[R\tPair(X):=\mathrm{Tot}_{R\cCoh(X)}(R\pi_* \FF)\]

of the perfect complex $R\pi_*\FF$ on $R\cCoh(X)$.
Moreover, the derived loop stack

\[\cL_{R\tPair(X),(F,s)}:=\Spec(\C)\times^h_{(F,s),R\tPair(X),(F,s)}\Spec(\C)\]

is an open substack of the fibre of the map between derived linear stacks

\[\mathrm{Tot}(R\Hom_X(F,F)) \xrightarrow{s} \mathrm{Tot}(R\Hom_X(\O_X,F))\]

at the point $s \in \Hom_X(\O_X,F)$. From the distinguished triangle

\[\xymatrix{R\Hom_X(I\udot,F)[-1] \ar[r] &R\Hom_X(F,F) \ar[r]^{s} & R\Hom_X(\O_X,F)}\]

the cotangent complex can be computed as

\[\TT_{R\tPair(X)}|_{(F,s)} \cong \TT_{\cL_{R\tPair(X),(F,s)}}|_{0}[1] \cong R\Hom_X(I\udot,F). \]

§.§ Geometric invariant theory

In this subsection, we construct moduli spaces of $\PT_q$ pairs as projective schemes via geometric invariant theory. Although we largely follow [135], our arguments differ since the underlying sheaves $F$ of $\PT_q$-stable pairs $(F,s)$ are not pure for $q=-1,0$. Our approach is inspired by [146].

Let $X$ be a smooth projective variety and $v \in H^*(X,\Q)$.
Then there exists a projective scheme $\cM$, an action of $\PGL_N$ on $\cM$, and $\SL_N$-equivariant ample line bundles $\cL^{(q)} \in \Pic^{\SL_N}(\cM)_{\Q}$, for each $q \in \{-1,0,1\}$, such that
\[\cPair^{(q)}(X,v) = [\cM^{st}(\cL^{(q)})/\PGL_N] = [\cM^{ss}(\cL^{(q)})/\PGL_N].\]

In particular, the moduli spaces $\cPair^{(q)}(X,v)$ are the GIT quotients $ \cM/\!/_{\cL^{(q)}}\SL_N$ and hence projective schemes.

The rest of this subsection is devoted to the proof of Theorem <ref>. By the boundedness result in Lemma <ref>, we have $\cPair^{(q)}(X,v) \subseteq \cPair(X,v)_m$ for sufficiently large $m$. Denoting the Hilbert polynomial determined by $v$ by $P_v(t)$, we then define

V:=\C^{P_v(m)}, \quad R_m := \Gamma(X,\O_X(m)).

We use the notation from subsection <ref> (e.g., the Quot scheme $\cQ$ in (<ref>), the universal family (<ref>), and the open subscheme $\cQ^{\circ}$ in (<ref>)).  Moreover, by Proposition <ref>, we have a presentation of the moduli stack as a global quotient stack,

\[\cPair(X,v)_m=[\PP_{\cQ^{\circ}}(R\pi_*\FF)/\PGL(V)].\]

We would like to use geometric invariant theory on $\PP_{\cQ^{\circ}}(R\pi_*\FF)$. Thus we need to consider a $\PGL(V)$-equivariant compactification of  $\PP_{\cQ^{\circ}}(R\pi_*\FF)$ and linearizations on it.

We define the parameter space $\cM$ as follows:

Define a projective scheme

\[\cM:=\overline{\PP_{\cQ^{\circ}}(R\pi_*\FF)} \subseteq \cQ \times \PP(V \otimes R_m^*)\]

as the closure of $\PP_{\cQ^{\circ}}(R\pi_*\FF)$ in $\cQ \times \PP(V \otimes R_m^*)$ with respect to the embedding

\begin{equation}\label{3}
\xymatrix{
\PP_{\cQ^{\circ}}(R\pi_*\FF) \ar@{^{(}->}[r] & \PP_{\cQ^{\circ}}(R\pi_*(\FF(m)) \otimes R_m^*)  & \\
& \PP_{\cQ^{\circ}}(\O_{\cQ^{\circ}}\otimes V\otimes R_m^*) \ar@{^{(}->}[u]_{\cong} \ar@{^{(}->}[r] & \cQ \times \PP(V\otimes R_m^*).
\end{equation}

Here the first map in (<ref>) is the closed embedding
induced by the canonical map

\[\Gamma(X,\O_X(m)) \otimes R\pi_*\FF \to R\pi_*(\FF(m)),\]

the second map in (<ref>) is an isomorphism induced by the canonical map (<ref>), and the last map in (<ref>) is the open embedding induced by $\cQ^{\circ} \hookrightarrow \cQ$.

The parameter space $\cM$ in this paper is larger than the parameter space of Le Potier in [135, 136]. Le Potier considered $\overline{\cQ}_{\mathrm{pure}} \times \PP(V\otimes R_m ^*)$ where $\overline{\cQ}_{\mathrm{pure}} \subseteq \cQ$ is the closure of the open subscheme $\cQ_{\mathrm{pure}}\subseteq \cQ$. The difference arises from the fact that we want to realize all three moduli spaces $\cPair^{(q)}(X,v)$ for $q=-1,0,1$, as GIT quotients of the same scheme $\mathcal{M}$ by the same group $\PGL(V)$ (with respect to different linearizations).

We consider the linearizations $\cL_{g(t),l}$ on $\cM$ defined as follows:

For a sufficiently large $l$, consider the standard Grothendieck embedding into the Grassmannian

\[\cQ \hookrightarrow \cG:={\curly Grass}(V\otimes R_{l-m}, P_v(l)) : (F,\rho) \mapsto (V\otimes R_{l-m} \twoheadrightarrow \Gamma(X,F(l))).\]

Fix a polynomial $g(t) \in \Q[t]$ with positive leading coefficient. Define the $\SL(V)$-equivariant very ample line bundle

\[\cL_{g(t),l} := \O_{\cG}(1)|_{\cQ}\boxtimes \O_{\PP(V\otimes R_m^*)}(g(l)) \in \Pic^{\SL(V)}(\cM)_\Q,\]

where $\O_{\cG}(1)$ is the polarization on $\cG$ induced by the standard Plücker embedding of $\cG$.

We describe the $\cL_{g(t),l}$-stable points via the Hilbert-Mumford criterion.

A point $(F,\rho,s)$
in $\cQ \times \PP(V\otimes R_m^*)$ is $\cL_{g(t),l}$-semistable if and only if the following two conditions are satisfied:

$(1)$ for all non-zero proper subspaces $V' \subseteq V$, we have

\begin{equation}\label{eq:ineq1}
\frac{P_{F'}(l)}{\dim(V')} \geq \frac{P_F(l) -g(l)}{\dim(V)}
\end{equation}

where $F':=\rho(V'\otimes \O_X(-m)) \subseteq F$, and

$(2)$ for all non-zero proper subspaces $V' \subseteq V$ with $s \in V' \otimes R_m^*$, we have

\begin{equation}\label{eq:ineq2}
\frac{P_{F'}(l) - g(l)}{\dim(V')} \geq \frac{P_F(l) -g(l)}{\dim(V)},
\end{equation}

where again $F':=\rho(V'\otimes \O_X(-m)) \subseteq F$.

Moreover, $(F,\rho,s)$ is $\cL_{g(t),l}$-stable if and only if the above two inequalities hold strictly.

This is standard by <cit.> (see also <cit.>). For completeness, we briefly go through the argument.

Note that $\cQ$ has a right $\SL(V)$-action and $\PP(V\otimes R_m^*)$ has a left $\SL(V)$-action. Here we will use the left-action convention, i.e., $\phi \cdot (F,\rho) = (F,\rho \circ \phi^{-1})$ for $\phi \in \SL(V)$.

Choose a 1-parameter torus $\lambda : \GG_m \to \SL(V)$. Equivalently, choose a weight decomposition
\[V = \bigoplus_{i \in \Z} V_i\]
such that $\sum_{i \in \Z } i \dim(V_i)=0$. Let $V_{\leq i}:= \bigoplus_{j\leq i}V_j$, $F_{\leq i}:=\rho(V_{\leq i}\otimes \O_X(-m))$, $F_i:=\frac{F_{\leq i}}{F_{\leq i-1}}$, and $\rho_i : V_i \otimes \O_X(-m) \twoheadrightarrow F_i$ be the induced surjective map. Let $s=\sum_i s_i$ such that $s_i \in V_i \otimes R_m ^*$ and let $i_{\max}$ be the maximal integer $i$ such that $s_i \neq 0$.

As in <cit.>, the limit point is
\[\lim_{t \to \infty} \lambda(t) \cdot (F,\rho,s) = (\bigoplus_i F_i, \bigoplus_i \rho_i, s_{i_{\mathrm{max}}}).\]
Since the fibre of the line bundle $\cL_{g(t),l}$ over a point $(F,\rho,s)$ is given as
\[\cL_{g(t),l}|_{(F,\rho,s)} = \det(H^0(X,F(l))) \otimes \C\langle s\rangle ^{\otimes-g(l)}\]

where $\C\langle s\rangle \subseteq V\otimes R_m^*$ is the subspace generated by $s$, we can easily compute the Hilbert-Mumford weight
\begin{align*} \mu^{\lambda^{-1}}_{\cL_{g(t),l}} (F,\rho,s) &= \sum_i \left(-i\cdot P_{F_i}(l) + i_{\mathrm{max}} \cdot g(l)\right)\\ &= \sum_i\left(  \left(P_{F_{\leq i}}(l) - \dim(\Gamma_{\leq i}) \cdot g(l) \right) -  \frac{\dim(V_{\leq i})}{\dim(V)} \cdot (P_F(l) -g(l)) \right) \end{align*}
where $\Gamma_{\leq i}:=\C\langle s \rangle \cap (V_{\leq i}\otimes R_m^*)$.

By <cit.>, the point $(F,\rho,s)$ is semistable if and only if
\[\frac{P_{F'}(l) - \dim(\Gamma') \cdot g(l)}{\dim(V')} \geq \frac{P_F(l) -g(l)}{\dim(V)}\]
for all proper subspaces $V' \subseteq V$, where $F':=\rho(V'\otimes \O_X(-m))$ and $\Gamma':=\C\langle s \rangle \cap (V'\otimes R_m^*)$. This completes the proof.

We will focus on the following three types of linearizations:

Recall that $P_v(t) \in \Q[t]$ is the Hilbert polynomial associated to $v \in H^*(X,\Q)$.

($\DT$) For any rational number $a>0$, we define a polynomial

\[g^{(-1)}(t):=P_v(t) + a\cdot t^2 .\]

($\PT_0$) For any rational number $0< \epsilon <1$, we define a polynomial

\[g^{(0)}(t):=P_v(t) -\epsilon \cdot t.\]

($\PT_1$) For any rational number $0< \delta <\tfrac12$, we define a polynomial

\[g^{(1)}(t):=P_v(t) - \delta \cdot t^2.\]

For each integer $q \in \{-1,0,1\}$, we define a linearization

\[\cL^{(q)}:= \cL_{g^{(q)}(t),l}\]

for sufficiently large $l$.

We list the linearizations previously considered in the literature [135, 136, 146].
* In [135], Le Potier considered $\O_{\cG}(c) \boxtimes \O_{\PP(V\otimes R_m^*)}(\frac{l^d}{d!})$ for $c>1$.
* In [136], Le Potier considered $\O_{\cG}(1+ \frac{\alpha(m)}{P_v(m)}) \boxtimes \O_{\PP(V\otimes R_m^*)}(P_v(l))$ for $\alpha(t) \in \Q[t]$.
* In [146], Stoppa-Thomas considered $\O_{\cG}(c) \boxtimes \O_{\PP(V\otimes R_m^*)}(l)$ for $c>0$ (for 1-dimensional pairs).
The linearizations $\cL_{g(t),l}$ in this paper are 2-dimensional versions of the linearizations in [146], which contain the linearizations in [135] as a special case.
However our parameter space is larger than the one in [135], as explained in Remark <ref>.

We now compare GIT stability for the line bundles $\cL^{(q)}$ in Definition <ref> and $\PT_q$-stability introduced in Definition <ref>.

There exists an integer $M$ which only depends on $X$ and $v$, and integers $L_m$ which depend on $X$, $v$, and $m$ such that for any $m \geq M$, $l \geq L_m$, and $(F,\rho,s) \in \cM$, we have the following:

$(1)$ $(F,\rho,s)$ is $\cL^{(q)}$-semistable $\implies$ $(F,\rho) \in \cQ^{\circ}$ and $(F,s)$ is $\PT_q$-stable;

$(2)$ $(F,\rho) \in \cQ^{\circ}$ and $(F,s)$ is $\PT_q$-stable $\implies$ $(F,\rho,s)$ is $\cL^{(q)}$-stable.

We will consider the three stability conditions separately. We start with $\PT_0$-stability.

(1) Assume that $(F,\rho,s)$ is $\cL^{(0)}$-semistable.

We first claim that the induced map

\[\Gamma(\rho(m)) : V \to \Gamma(X,F(m))\]

is injective. Let $V'$ be the kernel of the above map $\Gamma(\rho(m))$. Then the induced sheaf $F':=\rho(V'\otimes \O_X(-m)) \subseteq F$ is zero. If $V'\neq 0$, then the $\cL^{(0)}$-semistability of $(F,\rho,s)$ implies an inequality

\[\frac{0}{\dim(V')} \geq \frac{\epsilon \cdot l}{\dim(V)}\]

by Lemma <ref>(1). This leads to a contradiction for $l>0$. Hence $V'=0$.

We claim that $(F,s) \in \cPair(X,v)$. More precisely, there exists a section $\O_X \to F$ that fits into the commutative diagram

\[\xymatrix@C+2pc{
\C \ar[d]^s<EMAIL_ADDRESS>\Gamma(X,F) \ar@{^{(}->}[d] \\
V \otimes R_m^* \ar@{^{(}->}[r]^-{\Gamma(\rho(m))} & \Gamma(X,F(m)) \otimes R_m^*

determined by the dotted arrow. Indeed, consider the open subscheme

\[\cQ^{\star}:=\{(F,\rho) \in \cQ : V \xrightarrow{\Gamma(\rho(m))} \Gamma(X,F(m)) \text{ is injective}\}\]

of $\cQ$. Note that the restriction $\cM|_{\cQ^{\star}}$ of $\cM$ is the closure of $\PP_{\cQ^{\circ}}(R\pi_*\FF)$ in $\cQ^{\star}\times \PP(V\otimes R_m ^*)$. Hence we can form a commutative diagram

\[\xymatrix{
\cM|_{\cQ^{\star}}<EMAIL_ADDRESS>\ar@{^{(}->}[d] &  \PP_{\cQ^{\star}}(R\pi_*\FF) \ar@{^{(}->}[d]\\
\cQ^{\star}\otimes \PP(V\otimes R_m^*) \ar@{^{(}->}[r] & \PP_{\cQ^{\star}}(R\pi_*(\FF(m)) \otimes R_m^*)

of closed embeddings of projective schemes over $\cQ^{\star}$. By the result in the previous paragraph, we have $(F,\rho,s) \in \cM|_{\cQ^{\star}}$, which proves the claim.

Next, we claim that $Q:=\coker(\O_X \xrightarrow{s} F) \in \Coh_{\leq0}(X)$. Let

\[V':= V \cap \Gamma(X,\O_Z(m)) \quad\text{in}\quad \Gamma(X,F(m)),\]

where $\O_Z:=\im(\O_X \xrightarrow{s} F)$. Then the induced sheaf $F':=\rho(V'\otimes \O_X(-m)) \subseteq F$ is contained in $\O_Z \subseteq F$. Note that we have a commutative diagram

\[\xymatrix{
V'\otimes R_m^* \ar@{^{(}->}[r] \ar@{^{(}->}[d]  & \Gamma(X,\O_Z(m)) \otimes R_m^* \ar@{^{(}->}[d]^s & \Gamma(X,\O_Z) \ar@{_{(}->}[l] \ar@{^{(}->}[d]^s\\
V \otimes R_m^* \ar@{^{(}->}[r]^-{\rho(m)} & \Gamma(X,F(m)) \otimes R_m^* & \Gamma(X,F) \ar@{_{(}->}[l]

where the left square is cartesian. Hence we have $s \in V' \otimes R_m^*$. By Lemma <ref>(2), we have inequalities

\begin{equation}\label{eq:S2.7}
\frac{P_{\O_Z}(l) - g^{(0)}(l)}{\dim(V')} \geq\frac{P_{F'}(l) - g^{(0)}(l)}{\dim(V')} \geq \frac{\epsilon \cdot l }{\dim(V)},
\end{equation}

for sufficiently large $l$, where the two collections

\[\{\O_Z:=\im(\O_X \xrightarrow{s} F)\} \and \{F' := \im(V'\otimes \O_X(-m) \xrightarrow{\rho} F)\}\]

for fixed $X$, $v$, $m$ are bounded by <cit.>. Hence the set $\{P_{\O_Z}(t)\}$ of Hilbert polynomials is finite. Since the leading coefficient of the left-hand side of (<ref>), as a polynomial in $l$, should be positive, we can take $l$ sufficiently large and deduce $\alpha_2(\O_Z)=\alpha_2(F)$ and $\alpha_1(\O_Z)>\alpha_1(F) - \epsilon$.
Since $\epsilon<1$, we have $\alpha_1(\O_Z)=\alpha_1(F)$, and thus $Q$ is 0-dimensional.

We now show that $(F,\rho) \in \cQ^{\circ}$. By Lemma <ref>, the collection of coherent sheaves $F$ with $\ch(F)=v$ and $\coker(s) \in \Coh_{\leq 0}(X)$ is bounded for fixed $X,v$.
Hence we have $H^{>0}(X,F(m))=0$ for sufficiently large $m$, as desired.

We finally claim that $F \in \Coh_{\geq 1}(X)$. Let $T_0(F) \subseteq F$ be the 0-dimensional torsion sheaf and let

\[V' := \Gamma(X, T_0(F)(m)) \subseteq \Gamma(X,F(m))\xleftarrow{\cong} V,\]

where the last isomorphism follows from the result in the previous paragraph. Then the induced sheaf $F':=\rho(V' \otimes \O_X(-m)) \subseteq F$ is $T_0(F)$ since the 0-dimensional sheaf $T_0(F)$ is globally generated. If $V' \neq 0$, then we have an inequality

\[1= \frac{\chi(T_0(F))}{\chi(T_0(F)(m))} = \frac{\chi(F')}{\dim(V')}\geq \frac{\epsilon \cdot l }{\dim(V)}\]

by Lemma <ref>(1). This leads to contradiction for big enough $l$. Therefore $V'=0$ and $F'=T_0(F)=0$.

(2) Assume that $(F,\rho) \in \cQ^{\circ}$ and $(F,s)$ is $\PT_0$-stable. We will prove the $\cL^{(0)}$-stability of $(F,\rho,s)$ through Lemma <ref>. Choose a proper non-zero subspace $V' \subseteq V$ and let $F' := \rho(V' \otimes \O_X(-m)) \subseteq F$ be the induced subsheaf. Then we have a commutative square

\begin{equation}\label{10}
\xymatrix{
V' \ar@{^{(}->}[r] \ar@{^{(}->}[d] & \Gamma(X, F'(m)) \ar@{^{(}->}[d] \\
V \ar[r]^<<<<{\cong} & \Gamma(X,F(m)).
\end{equation}

We first prove the inequality (<ref>) in Lemma <ref>(1). Note that the collection

\[\{F' = \im(V' \otimes \O_X(-m) \xrightarrow{\rho} F) : (F,\rho) \in \cQ^{\circ} \and V' \subseteq V\}\]

for fixed $X,v,m$ is bounded by <cit.>. Hence the set $\{P_{F'}(t)\}$ of the corresponding Hilbert polynomials is finite. Since $F' \in \Coh_{\geq 1}(X)$ by $\PT_0$-stability and $\chi(F')$ has a lower bound, independent of the choices of $F$, $V'$, $l$, we have an inequality

\[\frac{P_{F'}(l)}{\dim(V')} > \frac{l + \chi(F')}{\dim(V)}> \frac{\epsilon \cdot l}{\dim(V)}\]

for sufficiently large $l$. This proves (<ref>).

We then prove the second inequality (<ref>) in Lemma <ref>(2) when $s \in V' \otimes R_m^*$. Indeed, consider the canonical commutative diagram

\begin{equation}\label{12}
\xymatrix{
\Gamma(X,F') \ar[r] \ar@{^{(}->}[d] & \Gamma(X,F) \ar[r] \ar@{^{(}->}[d] & \Gamma(X, F/F') \ar@{^{(}->}[d] \\
\Gamma(X,F'(m)) \otimes R_m^* \ar[r] & \Gamma(X,F(m)) \otimes R_m^* \ar[r] & \Gamma(X,F/F'(m)) \otimes R_m^*
\end{equation}

induced by the short exact sequence $0 \to F' \to F \to F/F' \to 0$. By (<ref>), we have $s \in \Gamma(X,F'(m)) \otimes R_m^*$.
By a diagram chasing argument, we can deduce that the left square in (<ref>) is cartesian. Hence $s \in \Gamma(X,F')$ and we have $\O_Z:=\im(s:\O_X \to F) \subseteq F'$ as subsheaves of $F$. Since $(F,s)$ is $\PT_0$-stable, we have $\alpha_2(F')=\alpha_2(F)$ and $\alpha_1(F')=\alpha_1(F)$. Since $\chi(F')$ has a lower bound, independent of the choices of $F$, $V'$, $l$, we have an inequality

\[\frac{\epsilon\cdot l + \chi(F') - \chi(F)}{\dim(V')} > \frac{\epsilon \cdot l}{\dim(V)}\]

for sufficiently large $l$. This proves (<ref>).

By [70, 135], the moduli spaces $\cPair^{(-1)}(X,v)$ and $\cPair^{(1)}(X,v)$ are projective schemes  (see Remark <ref>). We will use these results to prove Proposition <ref> for the $\DT$-stability and the $\PT_1$-stability.[Proposition <ref> for the $\PT_1$-stability is not a consequence of the results in [135], since the parameter space $\cM$ in this paper is larger than the parameter space in [135] (see Remark <ref>).]

(2) Assume that $(F,\rho) \in \cQ^{\circ}$ and $(F,s)$ is a $\DT$ pair. Choose a proper non-zero subspace $V' \subseteq V$ and let $F'=\rho(V'\otimes \O_X(-m)) \subseteq F$ be the induced subsheaf.

We first prove the inequality (<ref>). Note that the leading coefficient of the right-hand side of (<ref>), as a polynomial of $l$, is negative. On the other hand, the leading coefficient of the left-hand side of (<ref>) is always positive. Since the collection $\{F'\}$ for fixed $X$, $v$, $m$ is bounded by <cit.>, the set $\{P_{F'}(t)\}$ is finite. Hence we have (<ref>) for sufficiently large $l$.

We then prove the second inequality (<ref>) when $s \in V' \otimes R_m^*$. Indeed, we can form the commutative square (<ref>) and the commutative diagram (<ref>) where the left square is cartesian. Hence we have $s \in \Gamma(X,F')$.  Since $(F,s)$ is a $\DT$ pair, the coherent sheaf $F$ is generated by the section $s$. Therefore $F'=F$. We claim that $V'=V$, which then leads to a contradiction. Indeed, since $s \in V' \otimes R_m^*$, we have a commutative diagram

\[\xymatrix{
\C\cong\Gamma(X,\O_X) \ar[r]<EMAIL_ADDRESS>& \Gamma(X,F) \ar@{^{(}->}[d] \\
V' \otimes R_m^* \ar@{^{(}->}[r] & \Gamma(X,F(m)) \otimes R_m^*

for some dotted arrow. By adjunction, we can form a commutative diagram

\[\xymatrix{
R_m \ar[r]<EMAIL_ADDRESS>& \Gamma(X,F) \otimes R_m \ar[d] \\
V' \ar@{^{(}->}[r] & \Gamma(X,F(m)).

The collection $\{\I_{Z/X} : \ch(\O_Z)=v\}$ is bounded, so for $m$ sufficiently large (with lower bound only depending on $X,v$), we may assume that the map

\[R_m = \Gamma(X,\O_X(m)) \to \Gamma(X,F(m))\]

is surjective. Therefore, $V' \hookrightarrow V=\Gamma(X,F(m))$ is also surjective, which proves the claim.

(1) We first claim that

\[(F,\rho,s) \in \cM^{ss}(\cL^{(-1)})|_{\cQ^{\circ}} \implies  \text{$(F,s)$ is a $\DT$ pair}\]

for sufficiently large $m$ and $l$. Indeed, choose $(F,\rho,s) \in \cM^{ss}(\cL^{(-1)})|_{\cQ^{\circ}}$ and let $Z$ be the support of $s$ and $Q$ be the cokernel of $s$. Let $V'= \Gamma(X,\O_Z(m)) \subseteq V\cong\Gamma(X,F(m))$ and let $F'=\rho(V'\otimes \O_X(-m)) \subseteq \O_Z \subseteq F$. Then clearly $s\in V' \otimes R_m^*$. Thus we have the inequality (<ref>). Also note

\begin{equation} \label{eq:dimVdimV'}
-\frac{a}{\dim(V')} < - \frac{a}{\dim(V)}.
\end{equation}

As above, for fixed $X,v,m$ the collection of $\{F'\}$ is bounded, so there are finitely many $\{P_{F'}(t)\}$. Taking $l$ sufficiently large, (<ref>) contradicts (<ref>). Hence $V'=V$ and thus $F'=F$. Therefore $\O_Z=F$ and $(F,s)$ is a $\DT$ pair.

By the results in (2) and the previous paragraph, we have open embeddings

\[\left[\cM^{ss}|_{\cQ^{\circ}}(\cL^{(-1)})/\PGL(V)\right] \subseteq \cPair^{(-1)}(X,v) \subseteq \left[ \cM^{st}(\cL^{(-1)})/\PGL(V)\right] \]

which induces an open dense embedding

\[\cPair^{(-1)}(X,v) \hookrightarrow \cM^{st}(\cL^{(-1)})/\!/\PGL(V)\hookrightarrow \cM^{ss}(\cL^{(-1)})/\!/\PGL(V).\]

Since $\cPair^{(-1)}(X,v)$ is proper and $\cM^{ss}(\cL^{(-1)})/\!/\PGL(V)$ is separated, the above open embedding is an isomorphism. Hence $\cM^{st}(\cL^{(-1)})=\cM^{ss}(\cL^{(-1)})$ and $\cPair^{(-1)}(X,v)=[\cM^{st}(\cL^{(-1)})/\PGL(V)]$.

(2) Assume that $(F,\rho) \in \cQ^{\circ}$ and $(F,s)$ is a $\PT_1$ pair. Choose a proper non-zero subspace $V' \subseteq V$ and let $F'=\rho(V'\otimes \O_X(-m)) \subseteq F$ be the induced sheaf.

We first prove the inequality (<ref>). Indeed, since $F$ is pure, the subsheaf $F'$ is also 2-dimensional. Since $\delta<\frac12$ and $\dim(V')<\dim(V)$, we have an inequality

\[\frac{\alpha_2(F')}{2\dim(V')} > \frac{\delta}{\dim(V)},\]

between the leading coefficients of the both sides of (<ref>), as polynomials of $l$. For fixed $X,v,m$, the collection $\{F'\}$ is bounded by <cit.>, and hence the set $\{P_{F'}(t)\}$ is finite. Therefore we have (<ref>) for sufficiently large $l$.

We then prove the inequality (<ref>) when $s \in V' \otimes R_m^*$. Indeed, we can form the commutative square (<ref>). We can also form the commutative diagram (<ref>) where the left square is cartesian. Hence we have $s \in \Gamma(X,F')$.
Since $(F,s)$ is a $\PT_1$ pair, $\alpha_2(F)=\alpha_2(F')$. Since $\dim(V')< \dim(V)$, we have an inequality

\[\frac{\delta}{\dim(V')}> \frac{\delta}{\dim(V)}\]

between the leading coefficients of the both sides of (<ref>), as polynomials of $l$. Since the collection $\{F'\}$ is bounded, we have (<ref>) for sufficiently large $l$.

(1) We will show that

\begin{equation}\label{eq:S2.3}
(F,\rho,s) \in \cM^{ss}(\cL^{(1)})|_{\cQ^{\circ}} \implies  \text{$(F,s)$ is a $\PT_1$ pair}
\end{equation}

for sufficiently large $m$ and $l$. Choose $(F,\rho,s) \in \cM^{ss}(\cL^{(1)})|_{\cQ^{\circ}}$. Let $Q=\coker(s: \O_X \to F)$ be the cokernel and $\O_S = \im(s: \O_X \to F)$ the image.

We first claim that $Q \in \Coh_{\leq 1}(X).$ Let $V'= \Gamma(X,\O_S(m)) \subseteq \Gamma(X,F(m)) \cong V$ and let $F'=\rho(V'\otimes \O_X(-m)) \subseteq \O_S \subseteq F$. Then clearly $s\in V' \otimes R_m^*$ and in particular $V' \neq 0$. Hence we have inequality (<ref>). If $\alpha_2(F')< \alpha_2(F)$, then the leading term of the left-hand side of (<ref>) is negative since $\delta<\frac{1}{2}$. Since the collection $\{F'\}$ is bounded by <cit.> for fixed $X,v,m$, this leads to a contradiction for sufficiently large $l$. Hence $\alpha_2(F')=\alpha_2(F)$ and thus $\alpha_2(\O_S)=\alpha_2(F)$. This proves the claim.

We then claim that $F$ is pure. Consider the torsion subsheaf $TF\subseteq F$. Let $V'= \Gamma(X, TF(m)) \subseteq V$ and let $F'=\rho(V'\otimes \O_X(-m))\subseteq TF \subseteq F$. If $V'\neq 0$, then we have the inequality (<ref>). Since the degree of the left-hand side of (<ref>), as polynomial of $l$, is $\leq 1$ and the degree of the right-hand side of (<ref>) is 2, and the collection $\{F'\}$ is bounded, the inequality (<ref>) leads to a contradiction for sufficiently large $l$. Hence we deduce that

\begin{equation}\label{25}
\Gamma(X, TF(m))=0.
\end{equation}

Consider the short exact sequence

\[\xymatrix{ 0 \ar[r] & TF \ar[r] & F \ar[r] & F/TF \ar[r] & 0.}\]

Define $\O_Z = \mathrm{im}(\O_S \to F \to F/TF)$.
Then we obtain a short exact sequence

\begin{equation} \label{eqn:sesPT1pairR}
\xymatrix{ 0 \ar[r] & \O_Z \ar[r] & F/TF \ar[r] & R \ar[r] & 0,}
\end{equation}

where $R=\coker(\O_Z \to F/TF) \in \Coh_{\leq 1}(X)$. Then both $F/TF$ and $\O_Z$ are pure 2-dimensional sheaves such that $\alpha_2(F/TF)=\alpha_2(\O_Z)=\alpha_2(F)$. In particular, (<ref>) defines a $\PT_1$ pair. Moreover, $\alpha_1(F/TF)$ is bounded above by $\alpha_1(F)$ and bounded below $\alpha_1(\O_{Z})$, which is bounded below by a constant depending on $X,\alpha_2(F)$ but not on $m$ by <cit.>. In particular, $\alpha_1(F/TF)$ takes finitely many values.
By Proposition <ref>(2),  $\chi(F/TF)$ is bounded above by a constant depending on $X,v$ but not on $m$. Since $\chi(TF)=\chi(F)-\chi(F/TF)$, we have

\begin{equation}\label{26}
\chi(TF)\geq C(X,v)
\end{equation}

for a constant $C(X,v)$ depending on $X,v$ but independent of $m$. By the Riemann-Roch formula, we have

\[\chi(TF(m)) = r \cdot m + \chi(TF)\]

where $r = \alpha_1(F) - \alpha_1(F/TF) \geq 0$ since $TF \in \Coh_{\leq 1}(X)$. Moreover, $r \geq 0$ takes finitely many values independent of $m$. We deduce from (<ref>) and (<ref>) that, for all $F$ as above, we have $r=0$ for sufficiently large $m$, which means $TF$ is 0-dimensional. Hence $TF$ is globally generated and thus (<ref>) implies $TF=0$. Therefore $F$ is pure.

Combining the results in (2) and (<ref>), we have open embeddings

\[\left[\cM^{ss}|_{\cQ^{\circ}}(\cL^{(1)})/\PGL(V)\right] \subseteq \cPair^{(1)}(X,v) \subseteq \left[ \cM^{st}(\cL^{(1)})/\PGL(V)\right]\]

which induces an open dense embedding

\[\cPair^{(1)}(X,v) \hookrightarrow \cM^{st}(\cL^{(1)})/\!/\PGL(V)\hookrightarrow \cM^{ss}(\cL^{(1)})/\!/\PGL(V).\]

Since $\cPair^{(1)}(X,v)$ is proper and $\cM^{ss}(\cL^{(1)})/\!/\PGL(V)$ is separated, the above map is an isomorphism. Hence we have $\cM^{st}(\cL^{(1)})=\cM^{ss}(\cL^{(1)})$ and $\cPair^{(1)}(X,v)=[\cM^{st}(\cL^{(1)})/\PGL(V)]$.

Theorem <ref> is a direct corollary of Proposition <ref>.

Since the stacks

\[\cPair^{(q)}(X,v), \quad [\cM^{st}(X,\cL^{(q)})/\PGL(V)],\quad  [\cM^{ss}(X,\cL^{(q)})/\PGL(V)]\]

are open substacks of $\cPair(X,v)_m = [\cM|_{\cQ^{\circ}}/\PGL(V)] = [\PP_{\cQ^{\circ}}(R\pi_*\FF)/\PGL(V)]$, it suffices to compare the $\C$-points. Proposition <ref> completes the proof.

We end this subsection with the following question:

Does Theorem <ref> holds for $d$-dimensional $\PT_q$ pairs with $d>2$ as defined in Remark <ref>?

§.§ Moduli of complexes

In this subsection, we prove that the moduli spaces of $\PT_q$ pairs are open subschemes of the moduli space of perfect complexes. We basically follow the reasoning in [128], but the actual proof requires a finer argument.

Let $X$ be a smooth projective variety and let $v \in H^*(X,\Q)$.
Recall [86, 107] that there exists a moduli space

\[\cPerf(X,v)_{\O_X}^{\spl}\]

of simple perfect complexes $I\udot$ on $X$ with $\det(I\udot)\cong \O_X$ and $\ch(I\udot) = v$, as an algebraic space.

Let $X$ be a smooth projective variety of dimension $n\geq 4$, $v \in H^*(X,\Q)$, and $q\in \{-1,0,1\}$. Then we have an open embedding

\begin{equation}\label{eq:S2.1}
\cPair^{(q)}(X,v) \hookrightarrow \cPerf(X,v')_{\O_X}^{\mathrm{spl}} : (F,s) \mapsto I\udot:=[\O_X \to F]
\end{equation}

where $v':= 1-v \in H^*(X,\Q)$.

The rest of this subsection is devoted to the proof of Theorem <ref>. Throughout this subsection, we use the notation in Theorem <ref>.

To any $\PT_q$ pair $(F,s)$ on $X$, we can associate a perfect complex

\[I\udot:=[\O_X \xrightarrow{s} F]\]

with $\det(I\udot) \cong \O_X$. We first show that $I\udot$ is simple.

Given a $\PT_q$ pair $(F,s)$ on $X$, we have

\[\Ext^{<0}_X(I\udot,I\udot)=0 \and \Hom_X(I\udot,I\udot)=\C\]

for the associated perfect complex $I\udot:=[\O_X \xrightarrow{s} F]$.

The proof is analogous <cit.>. Consider the two canonical distinguished triangles

\[\xymatrix{I\udot \ar[r] & \O_X \ar[r] & F  } \and \xymatrix{\I_Z \ar[r] & I\udot \ar[r] & Q[-1]}\]

where $Z$ is the support of $F$ and $Q$ is the cokernel of $s$. We can form the following three induced distinguished triangles

\begin{align}
&\xymatrix{R\Hom_X(F,\O_X) \ar[r] & R\Hom_X(\O_X,\O_X) \ar[r] & R\Hom_X(I^\mdot,\O_X)},
\label{eq:S2.dt1}\\
&\xymatrix{R\Hom_X(Q,F)[1] \ar[r] & R\Hom_X(I^\mdot,F) \ar[r] & R\Hom_X(\I_Z,F)},
\label{eq:S2.dt2}\\
&\xymatrix{R\Hom_X(I^\mdot,I^\mdot) \ar[r] & R\Hom_X(I^\mdot,\O_X) \ar[r] &R\Hom_X(I^\mdot,F)}.
\label{eq:S2.dt3}
\end{align}

The distinguished triangle (<ref>) implies

\begin{equation}\label{Eq7}
\Hom_X(I^\mdot,\O_X)=\C \and \Ext^{<0}_X(I^\mdot,\O_X)=0
\end{equation}

since $\Ext^{\leq 1}_X(F,\O_X)=H^{\geq n-1}(X,F)^*=0$. On the other hand, the distinguished triangle (<ref>) implies

\begin{equation}\label{Eq8}
\Ext^{<0}_X(I^\mdot,F)=0
\end{equation}

since $\Hom_X(Q,F)=0$ by $\PT_q$-stability. Then the distinguished triangle (<ref>) and equations (<ref>) and (<ref>) prove that $I^\mdot$ is simple.

Since $\cPair^{(q)}(X,v)$ is a fine moduli space, there exists a universal family

\[\II\udot = [\O_{\cPair^{(q)}(X,v)\times X} \xrightarrow{} \FF],\]

which defines a map

\begin{equation}\label{eq:S2.2}
\cPair^{(q)}(X,v) \xrightarrow{\II\udot} \cPerf(X,v)_{\O_X}^{\spl}
\end{equation}

between the moduli spaces by Lemma <ref> below.

Let $f:\sX \to \sY$ be a smooth projective morphism of schemes. Let $\mathcal{F}$ be a $\sY$-flat coherent sheaf on $\sX$. If $\mathrm{codim}(\Supp(\mathcal{F}_y)) \geq 2$ for all $y \in \sY$, then $\det(\mathcal{F}) \cong \O_{\sX}$.

This follows directly from <cit.>.

We show that the above map (<ref>) satisfies the infinitesimal lifting property.

Let $B_0 \subseteq B$ be an nilpotent extension of 0-dimensional schemes with a unique closed point $b \in B_0$. Let $I\udot$ be a perfect complex on $X \times B$ with trivial determinant.
Let $i:X \times B_0 \hookrightarrow X\times B$ be the inclusion map.
If $I\udot|_{X\times B_0} \cong [\O_{X\times B_0} \xrightarrow{s_0} F_0]$ for a $B_0$-flat family $(F_0,s_0)$ of $\PT_q$ pairs over $B_0$, then there exists a $B$-flat family $(F,s)$ of $\PT_q$ pairs over $B$ such that $I\udot\ \cong [\O_{X\times B} \xrightarrow{s} F]$ and $(F,s)|_{X\times B_0}=(F_0,s_0)$.

We first take $q = 0, 1$. The (easier) case $q=-1$ is discussed at the end of the proof. Let $(F_b,s_b)$ be the fibre of $(F_0,s_0)$ over the point $b \in B_0 $. Since $F_b \in \Coh_{\geq1}(X)$, we have $\mathrm{depth}_{\O_{X,x}}((F_b)_x) \geq 1$ for all closed points $x \in \Supp(F_b)$.
By the Auslander-Buchsbaum formula, we have

\[\mathrm{pd}_{\O_{X,x}}((F_b)_x) \leq n-1\]

for all $x \in X$.
Then $I^{\mdot}|_{X\times\{b\}}\cong[\O_X \xrightarrow{s_b} F_b]$ has tor-amplitude $[-(n-2),1]$, and thus we have a quasi-isomorphism with a complex of locally free sheaves

\begin{equation} \label{eqn:resIb}
I^\mdot|_{X \times \{b\}}\cong[F^{-(n-2)}\to \cdots \to F^0 \to F^1].
\end{equation}

Since $X\times B$ is quasi-projective, $I^\mdot$ is quasi-isomorphic to a bounded complex of locally free sheaves. Using Nakayama's lemma, and the fact that a morphism between locally free sheaves is injective on the fibres if and only if it is injective with locally free cokernel, we see from (<ref>) that there exists a quasi-isomorphism with a complex of locally free sheaves

\[I^\mdot\cong[E^{-(n-2)}\to \cdots \to E^0 \to E^1].\]

Moreover since $I\udot$ has amplitude $[0,1]$ by <cit.>, we have

\[I^\mdot \cong [A \to E^1]\]

where $A=\coker(E^{-1} \to E^0)$ is $B$-flat.

We claim that the canonical map

\[h^0(I^\mdot)\to h^0(I^\mdot)^{**}\]

is injective. Indeed, let $K$ be the kernel of the above map $h^0(I^\mdot)\to h^0(I^\mdot)^{**}$. Note that $h^0(I^\mdot)$ is a vector bundle away from the support of $F_0$ by <cit.>.
Hence $\dim(K) \leq 2 < n$.

The support $W$ of $h^1(I^\mdot)$ is of dimension $\leq q$. This follows from the fact that $i^*h^1(I^\mdot)\cong h^1(I_0^\mdot)$
and Nakayama's lemma.
Away from $W$, the coherent sheaf $h^0(I^\mdot)$ is $B$-flat because $A$ is $B$-flat.
Since the ideal sheaf $h^0(I_b^\mdot)$ is torsion free, all non-zero subsheaves of $h^0(I^\mdot)|_{X \times B \setminus W}$
are $n$-dimensional by Lemma <ref> below. Therefore $K$ is (set-theoretically) supported in $W$. Thus we have $\dim(K)\leq  \dim(W) \leq q.$

Consider the local-to-global spectral sequence

\[H^i(X\times B,\ext^j_{X\times B}(A,K\otimes \omega_X))\Rightarrow \Ext^{i+j}_{X\times B}(A,K\otimes \omega_X).\]

$\dim(K\otimes \omega_X) \leq q \leq 1$
and there is a resolution $A\cong[E^{-(n-2)}\to \cdots \to E^0]$ of length $n-2$, we have

\[\Ext^n_{X\times B}(A,K\otimes \omega_X)=0.\]

By Serre duality $R\hom_{\pi}(K,A)\cong R\hom_{\pi}(A, K \otimes \omega_X))^{\vee}[-n],$ we have

\[\Hom_{X\times B}(K,A)^*\cong\Ext^n_{X\times B}(A,K\otimes \omega_X)=0,\]

where the dual is taken as a $\Gamma(B,\O_B)$-module.
Since $K \subseteq A$, we deduce $K=0$.
This proves the claim.

Since the rest of the proof is exactly the same as [128], we will only sketch the remaining arguments. By <cit.>, $h^0(I^\mdot)^{**}$ is a vector bundle away from the support of $Q=h^1(I^\mdot)$.
In the proof of <cit.>, it is also shown that a reflexive rank 1 sheaf which is locally free outside a subset of codimension $n-1 \geq 3$, is in fact locally free everywhere. Therefore $h^0(I^\mdot)^{**}$ is locally free.

Next, we have $\det(I^\mdot) \cong \O_{X \times B}$ and $W \ \subset \mathrm{Supp}(F_0)$, thus

h^0(I^\mdot)^{**}|_{(X\times B)\setminus \mathrm{Supp}(F_0)} \cong h^0(I\udot)|_{(X\times B)\setminus \mathrm{Supp}(F_0)} \cong \O_{(X\times B) \setminus \mathrm{Supp}(F_0)}.

Since $\mathrm{Supp}(F_0)$ has codimension $n-2\geq 2$, we find $h^0(I\udot)^{**}\cong \O_{X\times B}$ globally and $\I_Z \cong h^0(I^\mdot)$ is an ideal sheaf of a closed subscheme $Z \subseteq X \times B$.
The distinguished triangle

\[\xymatrix{\I_Z \ar[r] & I^\mdot \ar[r] & Q[-1]}\]

defines an element

\[\alpha\in \Ext^2_{X\times B}(Q,\I_Z)\cong\Ext^1_{X\times B}(Q,\O_Z)\]

since $\Ext^1_{X\times B}(Q,\O_{X\times B})=\Ext^2_{X\times B}(Q,\O_{X\times B})=0$ (recall $q \leq 1$ and $n \geq 4$). Hence we have a short exact sequence

\[\xymatrix{0 \ar[r] & \O_Z \ar[r] & F \ar[r] & Q \ar[r] & 0}\]

of coherent sheaves on $X\times B$. Therefore we have an isomorphism

\[I^\mdot \cong [\O_{X\times B} \xrightarrow{s} F]\]

of perfect complexes. Since $F$ and $F_0$ are 2-dimensional, $\Hom_{X\times B}(I^\mdot,\O_{X\times B}) \cong \Gamma(X\times B,\O_{X\times B}) \cong \Gamma(B,\O_B)$ and $\Hom_{X\times B_0}(I_0^\mdot,\O_{X\times B_0}) \cong \Gamma(B_0,\O_{B_0})$. As in [128], we therefore have $L i^*F \cong F_0$. Hence $F$ is flat over $B$ by <cit.>, which completes the proof.

For $q=-1$, $I^\mdot$ is only quasi-isomorphic to a bounded complex of locally free sheaves $[E^{-(n-1)} \to \cdots \to E^1]$. However, this time $W = \emptyset$, so $h^0(I^\mdot)\to h^0(I^\mdot)^{**}$ is obviously injective (Lemma <ref>). The rest of the proof proceeds as in the previous paragraph.

We need the following lemma to complete the proof of Proposition <ref>.

Let $(A,\m)$ be a local Artinian $\C$-algebra and let $X$ be a quasi-projective scheme. Consider an $A$-flat family $F$ of sheaves on $X\times \Spec(A)$ whose fibre $F_0=F \otimes_A A/\m$ is a pure sheaf. Then $F$ is also a pure sheaf, i.e., for all non-zero subsheaves $G \subseteq F$, we have $\dim(G)=\dim(F)$.

Since $A$ is Artinian, we may take a sequence of ideals <cit.>

\[0=\m_N\subset \cdots \subset\m_2\subset \m_1 =\m,\quad\text{such that}\quad\m_i/\m_{i+1} \cong A/\m\,.\]

We let $\m_0=A$. Suppose we have a subsheaf $G\subset F$ such that $\dim(G)<\dim(F)$. Since $F$ is flat over $A$, we have a short exact sequence

\begin{equation}\label{Eq1}
\xymatrix{0 \ar[r]& \m_{i+1}\cdot F \ar[r]& \m_i \cdot F \ar[r]& F_0 \ar[r]& 0}
\end{equation}

of sheaves on $X \times \Spec(A)$, where $\m_i \otimes_A F=\m_i\cdot F$. Consider the induced morphism

\[\xymatrix{
0 \ar[r] & G \cap \m\cdot F \ar[r] \ar@{^{(}->}[d] & G \ar[r] \ar@{^{(}->}[d] & \frac{G}{G \cap \m\cdot F} \ar[r] \ar@{^{(}->}[d] & 0 \\
0 \ar[r] & \m\cdot F \ar[r] & F \ar[r] & F_0 \ar[r] & 0}\]

of short exact sequences. Note that

\[\dim(\tfrac{G}{G\cap \m \cdot F}) \leq \dim(G)<\dim(F)=\dim(F_0).\]

Since $F_0$ is pure, we have $\frac{G}{G\cap \m \cdot F} = 0$. Therefore we deduce that

\[G \subseteq \m \cdot F.\]

Moreover, applying this argument to (<ref>), we deduce that

\[G \subseteq \m_i \cdot F \implies G \subseteq \m_{i+1} \cdot F.\]

Therefore we obtain $G \subseteq \m_N \cdot F = 0$ by induction on $i$.

Now we can simply deduce Theorem <ref> from Lemma <ref> and Proposition <ref> as follows:

By Lemma <ref>, the map

\[\cPair^{(q)}(X,v) \xrightarrow{\II\udot} \cPerf(X,v)_{\O_X}^{\spl}\]

in (<ref>) is injective on $\C$-points. Since $\C$ is algebraically closed, the above map is universally injective. In order to verify that the above map is formally smooth it is enough to check the lifting property along extensions of local Artinian $\C$-algebras <cit.> which is done in Proposition <ref>. Since $\cPair^{(q)}(X,v)$ is of finite type, the above map is étale. Moreover the above map is universally injective and étale, thus it is an open embedding (<cit.>).

The open embedding (<ref>) in Theorem <ref> induces a derived enhancement of the moduli space $\cPair^{(q)}(X,v)$ of $\PT_q$ pairs. The tangent complex at $(F,s) \in \cPair^{(q)}(X,v)$ is given as

\[\TT|_{(F,s)} = R\Hom_X(I\udot,I\udot)_0[1]\]

where $I\udot=[\O_X \xrightarrow{s} F]$. This derived structure (of perfect complexes) is different from the derived structure (of pairs) in Remark <ref>.

For $\PT_1$-stability, Theorem <ref> was independently proved by Gholampour-Jiang-Lo in <cit.> using <cit.>.

In the proof of Proposition <ref> we only used the fact that $T_0(F)=0$ and $Q\in \Coh_{\leq 1}(X)$, and the statement can be generalized as follows.
Let $\cPerf(X)_{\O_X}$ be the moduli stack of universally gluable perfect complexes $I\udot$ on $X$ with $\det(I\udot)\cong \O_X$ which is an algebraic stack locally finite type over $\mathbb{C}$ <cit.>. By the proof of Lemma <ref> we have an open substack

\[\cPair^{[0,1]}(X):= \Big\{(F,s) \in \cPair(X) : T_0(F)=0 \and Q \in \Coh_{\leq 1}(X) \Big\}\]

inside $\cPair(X)$. For $q=0,1$, we have

\[\xymatrix{
\cPair^{(q)}(X) \ar@{^{(}->}[r] \ar@{^{(}->}[rd] & \cPair^{[0,1]}(X) \ar@{^{(}->}[r] \ar[d]_\epsilon & \cPair(X)\,, \\
& \cPerf(X)_{\O_X}

where the horizontal morphisms are open embedding and the morphism $\epsilon$ is formally smooth by Proposition <ref>.
We claim that the map

\[\epsilon : \cPair^{[0,1]}(X) \to \cPerf(X)_{\O_X}\]

is an open embedding.
Indeed, it suffices to show that the induced map $\epsilon(\C)$ on the groupoids of $\C$-valued points is fully faithful. By the same argument as for Lemma <ref>, the induced map $\pi_0(\epsilon(\C))$ on the sets of isomorphism classes is injective. By (<ref>) (in fact by (<ref>)), we have an exact sequence

\[0\to \Ext_X^{-1}(I\udot,F)\to \Hom_X(I\udot,I\udot)_0\to \Ext^1_X(F,\O_X)=0\]

for $(F,s)\in\cPair^{[0,1]}(X)$ and hence $\epsilon(\C)$ preserves stabilizers. This proves the claim.

§.§ Polynomial Bridgeland stability

In this subsection, we prove that the $\PT_q$-stability conditions can be realized as Bayer's polynomial Bridgeland stability conditions [29, 11] on the derived category.
The results of this subsection, which are inspired by [59], are not used elsewhere in the paper. We use a different torsion pair compared to [59], which allows us to incorporate 2-dimensional (Gieseker) stable sheaves as well.

Let $X$ be a smooth projective variety of dimension $n \geq 4$. Let $D^b _{\mathrm{coh}}(X)$ be the bounded derived category of coherent sheaves on $X$.
We consider the perversity function

p : \{0,1,\ldots, n\} \to \Z, \quad p(0)=p(1)=p(2)=0, \quad p(3)=\ldots =p(n)=-1.

The perversity function defines a bounded $t$-structure on $D^b_{\mathrm{coh}}(X)$ [7] and this $t$-structure coincides with the one associated to the torsion pair

(\Coh_{\leq2} (X), \Coh_{\geq3} (X)),

<cit.>. We denote the heart by

\cA := \cA^p := \langle \, \Coh_{\leq2} (X), \Coh_{\geq3} (X) [1]\, \rangle\,.

The abelian category $\cA$ consists of the objects $E\udot \in D^{[-1,0]}_{\mathrm{coh}}(X)$ satisfying

\[h^{-1}(E^\mdot) \in \Coh_{\geq3}(X) \quad \textrm{and} \quad  h^0(E^\mdot) \in \Coh_{\leq 2}(X).\]

A sequence
$0\to A\udot\to E\udot\to B\udot \to 0$  is a short exact sequence in $\cA$ if and only if the associated sequence in $D^b_{\mathrm{coh}}(X)$ is a distinguished triangle. For any $E\udot\in \cA$, the following sequence is exact in $\cA$

\[0\to h^{-1}(E\udot)[1]\to E\udot\to h^0(E\udot)\to 0.\]

For any $\PT_q$ pair $I^\mdot$ on $X$, we have $I^\mdot[1] \in \cA$. Throughout this subsection, we fix $v\in H^*(X,\mathbb{Q})$ such that
$v_{\leq n-3} = 0$ and $v_{n-2}$ is non-zero effective.

Denote the semi-closed upper half plane by

\mathbb{H} = \{ z \in \C \, : \, z \in \R_{>0} \cdot e^{i \pi \phi(z)} \quad 0 < \phi(z) \leq 1 \},

where $\phi(z)$ denotes the phase of $z$. Denote by $\mathbb{H}^\circ \subset \mathbb{H}$ the open upper half plane. Recall [11] that a Bayer stability condition (also known as a polynomial stability condition) is determined by a stability vector, i.e., a sequence

\[\rho = (\rho_0,\ldots , \rho_n)\in (\C^*)^{n+1}\]

such that $\rho_d/\rho_{d+1} \in \mathbb{H}^\circ$ for $0\leq d \leq n-1$ and $(-1)^{p(d)}\rho_d\in\mathbb{H}$ for all $d$. We fix a polarization $H$ of $X$.
Define the central charge $Z\colon K^0(X)\to \C [x]$ as

\[Z(E^\mdot) (x) = \sum_{d=0}^n  x^d  \rho_d \int_X H^d \ch(E^\mdot)\td(X),\]

for any $E^\mdot \in K^0(X)$.
For $E\udot\in\cA$, let $\phi$ be the polynomial phase function, i.e.,

\[Z(E^\mdot)(x) \in \R_{>0} \cdot e^{\pi i \phi(E^\mdot)(x)}, \quad x\gg 0,\]

for some continuous function germ
$\phi$. We say $\phi<\psi$ if and only if $\phi(x)<\psi(x)$ for all $0\ll x<+\infty$, and similarly for $\phi \leq \psi$.

An object $E\udot$ in $\cA$ is called Bayer stable (resp. Bayer semi-stable) with respect to the stability vector $\rho$ if for all proper non-zero subobjects $D\udot\subset E\udot$, we have $\phi(D\udot)< \phi(E\udot)$ (resp. $\phi(D\udot)\leq \phi(E\udot)$).

Consider stability vectors $\rho^{(q)}$ for $q \in \{-1,0,1\}$ satisfying the following:

\begin{equation}\label{Eq.Bayerstabilityvectors}
\begin{tabular}{c|c}
$\rho^{(-1)}$  & $\phi(-\rho_n)>\phi(\rho_0) >\phi(\rho_1) >\phi(\rho_2)$ \\
\hline $\rho^{(0)}$  & $\phi(\rho_0)>\phi(-\rho_n) >\phi(\rho_1) >\phi(\rho_2)$\\
\hline $\rho^{(1)}$ & $\phi(\rho_0)>\phi(\rho_1)>\phi(-\rho_n) >\phi(\rho_2)$.
\end{tabular} }
\end{equation}

From the condition on the stability vector, we have

\begin{equation} \label{eqn:Bayerstabaux}
\phi(-\rho_3),\ldots, \phi(-\rho_{n-1}) > \phi(-\rho_n).
\end{equation}

The following proposition is a minor modification of <cit.>.

Fix $v \in H^*(X,\Q)$ such that $v_{\leq n-3}=0$ and $v_{n-2}\neq0$.
Let $E^\mdot \in D^b_{\mathrm{coh}}(X)$ with $\ch(E^{\mdot}) = v-1$ and $\det(E^\mdot) \cong \O_X$. Let $q\in\{-1,0,1\}$. Then  $E^\mdot$ is a $\rho^{(q)}$-semistable object of $\cA$ if and only if $E^\mdot \cong I^\mdot[1]$ for some $\PT_q$ pair $(F,s)$ where $I^\mdot=[\O_X \xrightarrow{s}F]$ is the associated complex. Moreover, $E^\mdot$ cannot be strictly $\rho^{(q)}$-semistable.

Let $E\udot$ be a $\rho^{(q)}$-semistable object of $\cA$ satisfying the above conditions. We first show that $h^{-1}(E\udot)$ is a rank one torsion free sheaf. Since $h^{0}(E\udot)\in \Coh_{\leq2}(X)$,  $h^{-1}(E\udot)$ has rank $1$ and $h^{-1}(E\udot)\in \Coh_{\geq 3}(X)$. Suppose $h^{-1}(E\udot)$ has a non-trivial torsion subsheaf. Let $T\subset h^{-1}(E\udot)$ be the maximal torsion subsheaf. Then $0\to T[1]\to h^{-1}(E\udot)[1]\to (h^{-1}(E\udot)/T)[1]\to 0$ is a short exact sequence in $\cA$ and hence $T[1]$ is a subobject of $E\udot$ in $\cA$. Then (<ref>) leads to a contradiction. Therefore $h^{-1}(E\udot)$ is a rank one torsion free sheaf with trivial determinant. By the argument in Proposition <ref>, we have $E\udot \cong I\udot[1]$, $I\udot=[\O_X\to F]$ for some 2-dimensional coherent sheaf $F$.
For $q=1$, we show that $F$ is pure and $h^0(E\udot)\in \Coh_{\leq 1}(X)$.  Let $T\subset F$ be the maximal torsion subsheaf of $F$. Then $0\to T\to I\udot[1]\to [\O_X\to F/T][1]\to 0$ is a short exact sequence in $\cA$ and hence $T$ is a subobject of $E\udot$.
By the stability vector (<ref>), we have $T=0$. Again by (<ref>) we have $h^0(E\udot)\in \Coh_{\leq1}(X)$. For $q=0$, similar argument shows that $F$ has no non-trivial 0-dimensional subsheaf and $h^0(E^\mdot)$ is 0-dimensional. For $q=-1$ we have $h^0(E^\mdot) = 0$.

Conversely, suppose $E^\mdot \cong I^\mdot[1]$ for some $\PT_q$ pair $(F,s)$ where $I^\mdot=[\O_X \xrightarrow{s}F]$. Then $E\udot\in\cA$. We prove $E\udot$ is a $\rho^{(q)}$-stable object.
Take any short exact sequence

0 \to A\udot \to E\udot \to B\udot \to 0

in $\cA$
with $A^\mdot, B^\mdot \neq 0$.
Consider its long exact sequence

0 \to h^{-1}(A^\mdot) \to h^{-1}(E^\mdot) \to h^{-1}(B^\mdot) \to h^{0}(A^\mdot) \to h^{0}(E^\mdot) \to h^{0}(B^\mdot) \to 0.

We distinguish three cases:

Case 1: $0 \neq h^{-1}(A^\mdot) \subsetneq h^{-1}(E^\mdot)$. Then $h^{-1}(B^\mdot)$ has rank 0. If $h^{-1}(B^\mdot) \neq 0$, then it has dimension in $\{3, \ldots, n-1\}$ so we obtain
$\phi(E^\mdot) < \phi(B^\mdot)$ from (<ref>). Suppose $h^{-1}(B^\mdot) = 0$, then $B^\mdot \cong h^0(B^\mdot)$ and since $h^0(E\udot)$ is at most 1-dimensional, $h^0(B\udot)$ has dimension $\leq 1$. For $q=1$, (<ref>) implies that $\phi(E^\mdot) < \phi(B^\mdot)$. For $q=0$, we have $h^0(B^\mdot)$ is 0-dimensional so we also find $\phi(E^\mdot) < \phi(B^\mdot)$. Finally, for $q=-1$ we have $h^0(B^\mdot) = 0$ contradicting the assumption $B^\mdot \neq 0$.

Case 2: $h^{-1}(A^\mdot) = h^{-1}(E^\mdot)$. Then $h^{-1}(B^\mdot) \subset h^0(A^\mdot)$ so $h^{-1}(B^\mdot) = 0$ and $B^\mdot \cong h^0(B^\mdot)$. Then $\phi(E^\mdot) < \phi(B^\mdot)$ follows as in Case 1.

Case 3: $h^{-1}(A\udot) = 0$. Then $A\udot\cong h^0(A\udot)$ has dimension $\leq 2$ and hence $\Hom(A^\mdot, E^\mdot) \cong \Hom(A^\mdot,F)$. Since $A^\mdot$ is a non-zero subobject of $E^\mdot$ and $(F,s)$ is a $\PT_q$ pair, we deduce that $A^\mdot \cong h^0(A^\mdot)$ is 2-dimensional. Then (<ref>) implies $\phi(A^\mdot) < \phi(E^\mdot)$.

We characterize rank zero $\rho^{(q)}$-semistable objects in $\cA$.

Let $q \in \{-1,0,1\}$ and $v \in H^*(X,\Q)$ such that $v_{\leq n-3}=0$ and $v_{n-2}\neq0$.
An object $E\udot \in \cA$ with $\ch(E\udot) = v$ is $\rho^{(q)}$-stable (resp. $\rho^{(q)}$-semistable) if and only if $E\udot \cong F$ for some Gieseker stable (resp. semistable) sheaf $F$ on $X$ with $\ch(F)=v$.

Let $E\udot\in\cA$ be a $\rho^{(q)}$-(semi)stable object with $\ch(E\udot) = v$. Since $h^{-1}(E\udot)\in\Coh_{\geq 3}(X)$, we have $h^{-1}(E\udot)=0$ and hence $E\udot\cong h^0(E\udot)$. Then $h^0(E\udot)$ is Gieseker (semi)stable <cit.>. Conversely, it is easy to show that any Gieseker (semi)stable sheaf $F$ on $X$ with $\ch(F)=v$ determines a $\rho^{(q)}$-(semi)stable object in $\cA$ <cit.>.

§.§ Relative version

We now generalize the main results in this section to the relative setting.

We first fix some notation.
$f : \cX \to \cB$
be a smooth projective morphism of schemes of relative dimension $n$.
Consider the algebraic de Rham complex

\[\Omega^\bullet_{\cX/\cB}:=\left[\O_\cX \xrightarrow{d_{DR}} \Omega^1_{\cX/\cB} \xrightarrow{d_{DR}} \Omega^2_{\cX/\cB} \xrightarrow{d_{DR}} \cdots \xrightarrow{d_{DR}} \Omega^n_{\cX/\cB}\right],\]

as in [69, 48]. Define the Hodge bundles as

\begin{equation}\label{Eq:HdgHbdl}
\cH^k_{DR}(\cX/\cB):=R^kf_*\left(\Omega_{\cX/\cB}^\bullet\right).
\end{equation}

When $\cB$ is an affine scheme, we write $H_{DR}^k(\cX/\cB):=\Gamma(\cB,\cH^k_{DR}(\cX/\cB)).$

Note that there is a a natural spectral sequence

\[E_1^{pq}=R^qf_*\Omega^p_{\cX/\cB} \Rightarrow \cH^{p+q}_{DR}(\cX/\cB)\]

given by the stupid truncations

\[\Omega^{\geq p}_{\cX/\cB}:= \left[0 \to \cdots \to 0 \to \Omega^p_{\cX/\cB} \xrightarrow{d_{DR}} \Omega^{p+1}_{\cX/\cB} \xrightarrow{d_{DR}}  \cdots \xrightarrow{d_{DR}} \Omega^n_{\cX/\cB}\right].\]

By [48], the spectral sequence degenerates at $E_1$. Hence we have a filtration

\[F^p\cH^{p+q}_{DR}(\cX/\cB):=\im\left(R^{p+q}f_*\big(\Omega^{\geq p}_{\cX/\cB}\big) \to R^{p+q}f_*\left(\Omega^{\bullet}_{\cX/\cB}\right)\right)\]

of the Hodge bundles $\cH^{p+q}_{DR}(\cX/\cB)$ such that

\begin{equation}\label{Eq:HdgFilt}
F^p\cH^{p+q}_{DR}(\cX/\cB)\cong R^{p+q}f_*\left(\Omega^{\geq p}_{\cX/\cB}\right)  \and \frac{F^p\cH^{p+q}_{DR}(\cX/\cB)}{F^{p+1}\cH^{p+q}_{DR}(\cX/\cB)} \cong R^qf_*\Omega^p_{\cX/\cB}.
\end{equation}

Moreover, the Hodge bundles $\cH^{p+q}_{DR}(\cX/\cB)$, the filtrations $F^p\cH^{p+q}_{DR}(\cX/\cB)$, and the quotients $R^qf_*\Omega^p_{\cX/\cB}$ are vector bundles and stable under base change, by <cit.>.

Consider the Gauss-Manin connection

\[\nabla : \cH^k_{DR}(\cX/\cB) \to \cH^k_{DR}(\cX/\cB) \otimes \Omega_\cB^1.\]

As in [91], the Gauss-Manin connection $\nabla$ can be defined via the canonical filtration of $\bigwedge^p \Omega^1_\cX$ given by the short exact sequence

\[\xymatrix{0 \ar[r] & f^*\Omega_\cB^1 \ar[r] & \Omega_\cX^1 \ar[r] & \Omega_{\cX/\cB}^1 \ar[r] & 0}.\]

Indeed, the first two terms of the filtration give us a map

\[\Omega^p_{\cX/\cB} \to \Omega^{p-1}_{\cX/\cB} \otimes f^*\Omega^1_{\cB}[1].\]

Then by gathering the above maps for all $p$ and taking the derived pushforward $R^qf_*$, we obtain a flat connection $\nabla$ called the Gauss-Manin connection.

Following [19], we define the notion of horizontal sections of Hodge bundles.

Let $f:\cX\to\cB$ be a smooth projective morphism of schemes of constant relative dimension.
We say that a section $\tv \in \Gamma(\cB,\cH^k_{DR}(\cX/\cB))$ is horizontal if

\[\nabla(\tv) =0 \in \Gamma(\cB, \cH^k_{DR}(\cX/\cB)\otimes \Omega^1_{\cB}).\]

Roughly speaking, the horizontal sections $\tv$ of $\cH^k_{DR}(\cX/\cB)$ are locally constant families of cohomology classes $\tv_b \in H^*(\cX_b,\C)$ on the fibres $\cX_b:=\cX\times_\cB\{b\}$. The following remark justifies this interpretation.

Let $f:\cX\to\cB$ be a smooth projective morphism of schemes of constant relative dimension.

* By considering the associated analytic spaces, we have

\[\cH_{DR}^k(\cX/\cB)^\an \cong R^kf^\an_*\C \otimes_{\C} \O_{\cB^\an} \and \nabla^\an \cong \id \otimes d_{\cB^\an}\]

when $\cB$ is smooth [91].
Thus a section $\tv$ of $\cH^k_{DR}(\cX/\cB)$ is horizontal if and only if
$\tv^\an = v\otimes 1 $
for some $v \in \Gamma(\cB, R^k f^\an_*\C)$.

* Firstly, assume that $\cB= \Spec(A)$ for a local Artinian $\C$-algebra $A$. Then we have isomorphisms

\[H^k_{DR}(\cX/\cB) \cong H^k_{DR}(X)\otimes_\C A\]

by <cit.>, where $X$ is the fibre of $f:\cX\to\cB$ over the unique closed point and $H^k_{DR}(X):=H^k_{DR}(X/\Spec(\C))$.
Then the horizontal sections are exactly the sections of the form

\[v\otimes 1 \in H^k_{DR}(X)\otimes_\C A\]

for some $v \in H^k_{DR}(X) \cong H^k(X,\C)$.

* Secondly, for general $\cB$, a section $\tv \in \Gamma(\cB,\cH^k_{DR}(\cX/\cB))$ is horizontal if and only if its pullbacks

\[\tv|_{\cA} \in \Gamma(\cA, \cH^k_{DR}(\cX_\cA/\cA))\]

are horizontal for all morphisms $\cA=\Spec(A) \to \cB$ from local Artinian $\C$-algebras $A$. Here $\cX_\cA:=\cX\times_\cB \cA$.
This can be shown by the Krull intersection theorem.

We have a relative moduli space of $\PT_q$ pairs on the fibres of $f: \cX \to \cB$.

Let $\cB$ be a quasi-projective scheme and $f: \cX \to \cB$ a smooth projective morphism of relative dimension $n$. Let $\tv \in \bigoplus_p \Gamma(\cB,\cH^{2p}_{DR}(\cX/\cB))$ be a horizontal section
and $q\in \{-1,0,1\}$.
Let $\cPair^{(q)}(\cX/\cB,\tv)$ be the moduli stack consisting of triples $(b,F,s)$ of a morphism $b: T \to \cB$, a $T$-flat coherent sheaf $F$ on $\cX_T:=\cX\times_{\cB} T$, and a section $s\in \Gamma(\cX_T,F)$ such that for all $t \in T(\C)$ the pair $(F_t,s_t)$ is a $\PT_q$ pair with $\ch(F_t)=\tv_t \in H^*(\cX_t,\C)$. Then we have the following properties:

$(1)$ The canonical map

\[\cPair^{(q)}(\cX/\cB,\tv) \to \cB, \quad (b,F,s) \mapsto b\]

is a projective morphism of schemes.

$(2)$ If $n \geq 4$, then the canonical map

\[\cPair^{(q)}(\cX/\cB,\tv) \to \cPerf(\cX/\cB)_{\O_{\cX}}^\spl, \quad (b,F,s) \mapsto (b, [\O_{\cX_T} \xrightarrow{s} F])\]

is an open embedding.

Fix a polynomial $P(t) \in \Q[t]$ and let $\cPair^{(q)}(\cX/\cB,P)$ be the moduli stack of $\PT_q$ pairs on the fibres of $f:\cX\to\cB$ with Hilbert polynomial $P(t)$ with respect to some relative very ample line bundle.
Note that we can find a closed embedding $\cX \hookrightarrow \PP^N_{\cB}$ over $\cB$ since $f:\cX\to\cB$ is projective and $\cB$ is quasi-projective.
Consider the canonical maps

\[\cPair^{(q)}(\cX/\cB,P) \hookrightarrow \cPair^{(q)}(\PP^N_{\cB}/\cB,P) \xrightarrow{\cong}\cPair^{(q)}(\PP^N,P)\times \cB\]

where the first map is a closed embedding, the second map is an isomorphism, and the last term is a projective scheme over $\cB$ by Theorem <ref>.
Indeed, as a functor over $\cPair^{(q)}(\PP^N_{\cB}/\cB,P)$, $\cPair^{(q)}(\cX/\cB,P)$ is the zero locus of the canonical map

\[\I_{\cX/\PP^N_{\cB}} \otimes \FF \to \FF\]

where $\I_{\cX/\PP^N_{\cB}}$ is the ideal sheaf of $\cX\subseteq \PP^N_{\cB}$ and $(\FF,s)$ is the universal pair on $\cPair^{(q)}(\PP^N_{\cB}/\cB,P)\times_{\cB} \PP^N_{\cB}$.
By <cit.>, the map $\I_{\cX/\PP^N_{\cB}} \otimes \FF \to \FF$ can be represented by the abelian cone of a coherent sheaf on $\cPair^{(q)}(\PP^N_{\cB}/\cB,P)$ since $\FF$ is flat.
Hence the desired map is a closed embedding since the abelian cone is separated over the base. Therefore we have (1) since $\cPair^{(q)}(\cX/\cB,\tv)$ is an open and closed subscheme of the projective scheme $\cPair^{(q)}(\cX/\cB,P)$ (for appropriate $P$). This proves (1).

All the arguments in subsection <ref> also works in the relative setting. This proves (2).

We end this subsection with a technical proposition stating that the relative Chern character of a perfect complex is horizontal. This fact will be used several times in section <ref> and <ref>.

Let $f:\cX \to\cB$ be a smooth projective morphism of schemes of constant relative dimension.
Let $I\udot$ be a perfect complex on $\cX$. Let

\[\At_{\cX/\cB}(I\udot) : I\udot \to I\udot \otimes \Omega^1_{\cX/\cB}[1]\]

be the (relative) Atiyah class of $I\udot$.

\[\At_{\cX/\cB}^p(I\udot) :=\At_{\cX/\cB}(I\udot)\wedge\cdots\wedge\At_{\cX/\cB}(I\udot) : I\udot \to I\udot \otimes \Omega_{\cX/\cB}^p[p]\]

be the induced map.
We define the (relative) Chern character $\ch_p(I\udot)$ of $I\udot$ as

\begin{equation}\label{Eq:ch}
H^p(\cX,\Omega^p_{\cX/\cB}) \to \Gamma(\cB,R^pf_*\Omega^p_{\cX/\cB}), \quad \frac{1}{p!}\tr(\At_{\cX/\cB}^p(I\udot)) \mapsto \ch_p(I\udot).
\end{equation}

The Chern character $\ch_p(I\udot)$ in (<ref>) is horizontal. More precisely, we have the following:

Let $f:\cX\to\cB$ be a smooth projective morphism of schemes of constant relative dimension.
Let $I\udot$ be a perfect complex on $\cX$.
Then there exists a section

\[\widetilde{\ch_p}(I\udot) \in \Gamma(\cB,F^p\cH_{DR}^{2p}(\cX/\cB))\]

satisfying the following properties:

$(1)$ Under the projection map
$F^p\cH^{2p}_{DR}(\cX/\cB) \twoheadrightarrow R^pf_*\Omega^p_{\cX/\cB}$ given by (<ref>),
the section $\widetilde{\ch_p}(I\udot)$ maps to the Chern character $\ch_p(I\udot)$ in (<ref>).

$(2)$ Under the inclusion map $F^p\cH_{DR}^{2p}(\cX/\cB) \hookrightarrow \cH^{2p}_{DR}(\cX/\cB)$ given by (<ref>),
the section $\widetilde{\ch_p}(I\udot)$ maps to a horizontal section.

$(3)$ Under the restriction map $\Gamma(\cB,\cH_{DR}^{2p}(\cX/\cB)) \to H_{DR}^{2p}(\cX_b/\Spec(\C))$ for any $\C$-point $b \in \cB$, the section $\widetilde{\ch_p}(I\udot)$ maps to the usual Chern character $\ch_p(I\udot_b) \in H^{2p}(\cX_b,\C)$.

The section $\widetilde{\ch_p}(I\udot)$ is uniquely determined by properties $(2)$ and $(3)$.

We note that an analogous statement for the fundamental class $[\cZ]$ of a local complete intersection $\cZ \subseteq \cX$ was shown in <cit.>.

We briefly summarize how we prove Proposition <ref>.