In this section we prove Theorem 5.8 by realizing the moduli of Higgs triples as a certain moduli of sheaves on a surface.
Let X be a scheme over \(\mathbb {k}\), not necessarily smooth. Pick a line bundle L over X, and consider the projectivization \(S=\mathbb {P}_X(L\oplus \mathcal {O}_X)\) of its total space \({\text {Tot }}L\). Denote the complement of \({\text {Tot }}L\) in S by D; let also \(i{:}\,D\hookrightarrow S\) be the natural embedding, and \(\pi {:}\,S\rightarrow X\) the natural projection. Note that by definition of S and D we have \(R\pi _*\mathcal {O}(D)=\pi _*\mathcal {O}(D)=\mathcal {O}_X\oplus L^\vee \), and \(\pi \) induces an isomorphism \(D\simeq X\).
Let \(\mathcal T=\mathcal {O}_S(D)\oplus \mathcal {O}_S\), and consider the sheaf of \(\mathcal {O}_X\)-algebras \(\pi _*\mathcal Hom(\mathcal T,\mathcal T)\). We can write it as a matrix algebra over X; the opposite algebra, which we denote by \(\mathcal A\), is then obtained by transposition:
$$\begin{aligned} \pi _*\mathcal Hom(\mathcal T,\mathcal T)=\begin{pmatrix} \mathcal {O}&{}\quad \mathcal {O}\oplus L^\vee \\ 0 &{}\quad \mathcal {O}\end{pmatrix}, \qquad \mathcal A=\begin{pmatrix} \mathcal {O}&{}\quad 0 \\ \mathcal {O}\oplus L^\vee &{}\quad \mathcal {O}\end{pmatrix}. \end{aligned}$$
Note that \(\mathcal A\) can be seen as a twisted path algebra of the following quiver:
A left \(\mathcal A\)-module \(\mathcal V\) is then determined by a quadruple \((\mathcal V_1,\mathcal V_2, \varphi _0,\varphi _1)\), where \(\mathcal V_1,\mathcal V_2\in {\text {Coh }}X\), \(\varphi _0\in {\text {Hom }}(\mathcal V_1,\mathcal V_2)\), and \(\varphi _1\in {\text {Hom }}(\mathcal V_1\otimes L^\vee ,\mathcal V_2)\).
For any coherent sheaf \(E\in {\text {Coh }}S\), the Hom-sheaf \(\mathcal Hom(\mathcal T, E)=\pi _*(\mathcal T^\vee \otimes E)\) is naturally a left \(\mathcal A\)-module, given by the quadruple \((\pi _*E(-D),\pi _*E,\varphi _0,\varphi _1)\), where \((\varphi _0,\varphi _1)\) is the natural composition
$$\begin{aligned} \pi _*E(-D)\otimes (\mathcal {O}\oplus L^\vee )=\pi _*E(-D)\otimes \pi _*\mathcal {O}(D)\rightarrow \pi _*E. \end{aligned}$$
Since \(\mathcal T\) is an \((\mathcal A^{op},\mathcal {O}_S)\)-bimodule, we have a pair of adjoint functors
As a left module over itself, \(\mathcal A\) can be decomposed as a direct sum \(\mathcal P_1\oplus \mathcal P_2\), where
are the left \(\mathcal A\)-modules defined in Sect. 6.
The following proposition should be known to experts (for example, see remark at the end of [3]), but we include the proof for completeness.
Proposition 7.1
The pair of functors (28) establishes an equivalence of triangulated categories.
Proof
The proof is based on Beĭlinson’s lemma [3]. For any \(E\in {\text {Coh }}S\), there exists \(n>0\) such that E(nD) has no higher cohomology, and the counit map \(\pi ^*\pi _*E(nD)\rightarrow E(nD)\) is surjective. By the seesaw principle [34, Corollary 5.6], the kernel of this map has the form \(\pi ^*(\mathcal N)(-D)\), where \(\mathcal N\in {\text {Coh }}X\). Thus E admits a resolution of the form
$$\begin{aligned} 0\rightarrow \pi ^*(\mathcal N_2)(-(n+1)D)\rightarrow \pi ^*(\mathcal N_1)(-nD)\rightarrow E\rightarrow 0, \end{aligned}$$
where \(\mathcal N_1,\mathcal N_2\in {\text {Coh }}X\). Taking into account short exact sequences
$$\begin{aligned} 0\rightarrow \mathcal {O}((n-1)D) \rightarrow \mathcal {O}(nD)\oplus \mathcal {O}(nD)\rightarrow \mathcal {O}((n+1)D)\rightarrow 0, \end{aligned}$$
we see that as a triangulated category, \(\mathcal D^b({\text {Coh }}S)\) is generated by \({\text {Coh }}X\) and \(\mathcal T=\mathcal {O}_S(D)\oplus \mathcal {O}_S\). Similarly, \(\mathcal D^b(\mathcal A\text {-mod})\) is generated by \({\text {Coh }}X\) and \(\mathcal A=\mathcal P_1\oplus \mathcal P_2\) as a triangulated category by Corollary 6.3.
We have \(R\pi _*\mathcal Hom(\mathcal T,\mathcal {O}(D))=\mathcal P_1\), \(R\pi _*\mathcal Hom(\mathcal T,\mathcal {O})=\mathcal P_2\). Using Theorem 6.4, it is easy to check the following isomorphisms:
$$\begin{aligned}&R {\text {Hom }}(\pi ^*\mathcal E,\pi ^*\mathcal F)\simeq R {\text {Hom }}(\mathcal E,\mathcal F)\simeq R {\text {Hom }}(\mathcal E\otimes \mathcal P_1,\mathcal F\otimes \mathcal P_1),\\&R {\text {Hom }}(\pi ^*\mathcal E(D),\pi ^*\mathcal F)\simeq 0\simeq R {\text {Hom }}(\mathcal E\otimes \mathcal P_1,\mathcal F\otimes \mathcal P_2),\\&R {\text {Hom }}(\pi ^*\mathcal E,\pi ^*\mathcal F(D))\simeq R {\text {Hom }}(\mathcal E,\mathcal F\otimes (\mathcal {O}\oplus L^\vee ))\simeq R {\text {Hom }}(\mathcal E\otimes \mathcal P_2,\mathcal F\otimes \mathcal P_1),\\&R {\text {Hom }}(\pi ^*\mathcal E(D),\pi ^*\mathcal F(D))\simeq R {\text {Hom }}(\mathcal E,\mathcal F)\simeq R {\text {Hom }}(\mathcal E\otimes \mathcal P_2,\mathcal F\otimes \mathcal P_2). \end{aligned}$$
Applying Beĭlinson’s lemma, we conclude that the functor \(R\mathcal Hom_X(\mathcal T,-)\) is an equivalence of triangulated categories. Moreover, since the functor \(-\otimes ^L_{\mathcal A}\mathcal T\) is its left adjoint, it provides the inverse equivalence. \(\square \)
Let us apply this proposition to \(X=T\times C\), \(L=\mathcal {O}_T\boxtimes \omega _C\). Combining it with Corollary 6.6, we obtain an equivalence of groupoids
$$\begin{aligned} \Theta {:}\,\underline{\mathcal D^b({\text {Coh }}(T\times \mathbb {P}_C(\omega \oplus \mathcal {O})))}\simeq \underline{(Lp^*\mathcal A)\text {-}mod_{\mathcal D}}, \end{aligned}$$
(29)
where \(p{:}\,T\times C\rightarrow C\) is the projection. Moreover, this equivalence commutes with base change in T whenever the latter preserves bounded derived categories, e.g. for flat maps \(T'\rightarrow T\).
Remark 7.2
It would be desirable to express this as an equivalence of presheaves in groupoids. The problem is that groupoids on both sides of (29) are not functorial in T. Namely, boundedness of complexes is not preserved under pullbacks along general maps \(T'\rightarrow T\). Nevertheless, in the sequel we are only concerned with certain subgroupoids on both sides, see Proposition 7.7. Their objects will satisfy flatness condition over T, and therefor will be preserved under arbitrary base change, forming presheaves. We will thus abuse the notation for convenience, and say that the two sides of (29) form presheaves \(\underline{\mathcal D^b({\text {Coh }}S)}\) and \(\underline{\mathcal A\text {-}mod_{\mathcal D}}\) respectively.
From now on, let \(X=C\), so that \(S=\mathbb {P}_C(\omega \oplus \mathcal {O})\) compactifies the cotangent bundle \(T^*C\). Let us recall some properties of sheaves on S; we will closely follow the exposition in [32, Section 2]. The Neron–Severi group of S is given by \({\text {NS }}(S)=H^2(S,\mathbb {Z})=\mathbb {Z}D\oplus \mathbb {Z}f\), where f is the class of a fiber of \(\pi {:}\,S\rightarrow C\). Thus, for any coherent sheaf E on S we will write the first Chern class \(c_1(E)\) as a linear combination \(c_{1,D}(E)D+c_{1,f}(E)f\). The product in \({\text {NS }}(S)\) is determined by the following equalities:
$$\begin{aligned} f^2=0,\quad Df=1,\quad D^2=2-2g, \end{aligned}$$
where the last one follows from the fact that \(\mathcal {O}_S(D)|_D\simeq \omega ^{-1}\). Moreover, the canonical divisor of S is \(K_S=-2D\). We will write elements of \(H^{even}(S,\mathbb {Z})\) as triples \((a,b,c)\in \mathbb {Z}\oplus {\text {NS }}(S)\oplus \mathbb {Z}\); the same applies to \(H^{even}(C,\mathbb {Z})\). In this fashion, Todd classes of S and C are respectively given by
$$\begin{aligned} {\text {td }}S=(1,D,1-g),\quad {\text {td }}C=(1,1-g), \end{aligned}$$
and the pushforward along \(\pi \) in cohomology is given by
$$\begin{aligned} \pi _*(a,b_D D+b_f f,c)=(b_D,c). \end{aligned}$$
Given a sheaf \(E\in {\text {Coh }}S\), the Chern character of its derived pushforward \(R\pi _*E\) can be computed using Grothendieck–Riemann–Roch theorem. Namely, let \(a=c_{1,D}(E)\), \(b=c_{1,f}(E)\), \(r={\text {rk }}E\), and recall that
$$\begin{aligned}{\text {ch }}(E)=\left( r,c_1(E),\frac{c_1(E)^2-2c_2(E)}{2}\right) =(r,aD+bf,a^2(1-g)+ab-c_2(E)).\end{aligned}$$
We have:
$$\begin{aligned}&({\text {rk }}(R\pi _*E),\text { }c_1(R\pi _*E)+(1-g){\text {rk }}(R\pi _*E))={\text {ch }}(R\pi _*E){\text {td }}C=\pi _*({\text {ch }}E{\text {td }}S)\\&\quad =\pi _*((r,aD+bf,{\text {ch }}_2(E))(1,D,1-g))\\&\quad =(a+r,(r+2a)(1-g)+b+{\text {ch }}_2(E)). \end{aligned}$$
The result of this computation can be rewritten as follows:
$$\begin{aligned} {\text {rk }}(R\pi _*E)=a+r,\quad c_1(R\pi _*E)=a(1-g)+b+{\text {ch }}_2(E). \end{aligned}$$
(30)
For any nef divisor H on S, we can define a notion of H-semistability for sheaves on S. One example of nef divisor is given by f. Instead of giving general definitions, we will use the following characterization of f-semistable sheaves:
Lemma 7.3
([32, Lemma 4.3]) A torsion-free sheaf E on S is f-semistable if and only if its generic fiber over C is isomorphic to \(\mathcal {O}_{\mathbb {P}^1}(l)^{\oplus m}\) for some \(l\in \mathbb {Z}\), \(m\in \mathbb {N}\).
Lemma 7.4
For a torsion-free f-semistable sheaf E, the following numerical conditions are equivalent:
$$\begin{aligned} \mathrm{1.}~\pi _*E=0\,\text {and}\,{\text {rk }}R^1\pi _*E=0,\qquad \qquad \mathrm{2.}~{\text {rk }}E=-c_{1,D}(E),\quad \mathrm{3.}~l=-1. \end{aligned}$$
If these conditions are fulfilled, we further have \(H^0(E)=H^2(E)=0\), and \(H^1(E)=H^0(R^1\pi _*E)\).
Proof
Let us first prove the equivalence.
\(1\Rightarrow 2\): follows from the first formula in (30);
\(2\Rightarrow 3\): rank is a generic invariant, therefore we have
$$\begin{aligned} 0={\text {rk }}R\pi _*E=m\left( h^0(\mathcal {O}_{\mathbb {P}^1}(l))-h^1(\mathcal {O}_{\mathbb {P}^1}(l))\right) =m(l+1). \end{aligned}$$
Since m is a positive number, this implies that \(l=-1\).
\(3\Rightarrow 1\): since \(R\Gamma (\mathcal {O}_{\mathbb {P}^1}(-1))=0\), both \(\pi _*E\) and \(R^1\pi _*E\) have rank 0. Furthermore, let \(\mathcal T\subset \pi _*E\) be a torsion subsheaf. By adjunction, we exhibit a map \(\pi ^*\mathcal T\rightarrow E\) from a torsion sheaf to a torsion-free sheaf. It is a zero map if and only if \(\mathcal T=0\); thus \(\pi _*E\) is locally free. We conclude that \(\pi _*E=0\).
In order to prove the second statement, recall that we have Leray spectral sequence
$$\begin{aligned} H^i(C,R^j\pi _*E)\Rightarrow H^{i+j}(S,E). \end{aligned}$$
Since \(R^j\pi _*E=0\) for \(j\ne 1\), it degenerates to the equality \(H^{i+1}(E)=H^i(R^1\pi _*E)\). Finally, \(R^1\pi _*E\) is a torsion sheaf, so that \(H^i(E)\) is non-zero only for \(i=1\). \(\square \)
We will also need the following computation:
Lemma 7.5
Let \(\varepsilon {:}\,\pi ^*\pi _*\mathcal {O}(D)\rightarrow \mathcal {O}(D)\) be the natural counit map. Then \({\text {Ker }}\varepsilon \simeq \pi ^*\omega ^\vee (-D)\).
Proof
Let us denote \(K={\text {Ker }}\varepsilon \). Since \(\varepsilon \) is surjective and becomes an isomorphism after applying \(\pi _*\), we have \(R\pi _*K=0\). This means that at each point \(c\in C\) the fiber \(K_c\) is isomorphic to a direct sum of several copies of \(\mathcal {O}_{\mathbb {P}^1}(-1)\) [34, Corollary 5.4]. In particular, \(K(D)_c\) is trivial at each point c, and thus the natural map \(\pi ^*\pi _*(K(D))\rightarrow K(D)\) is an isomorphism. Consider the following short exact sequence:
$$\begin{aligned} 0\rightarrow K(D)\rightarrow \left( \pi ^*\pi _*\mathcal {O}(D)\right) \otimes \mathcal {O}(D)\rightarrow \mathcal {O}(2D)\rightarrow 0. \end{aligned}$$
Note that all these sheaves have globally generated fibers over C. Therefore, after applying \(\pi _*\) we obtain
$$\begin{aligned} \pi _*(K(D))\simeq {\text {Ker }}\left( \pi _*\mathcal {O}(D) \otimes \pi _*\mathcal {O}(D)\rightarrow \pi _*\mathcal {O}(2D) \right) . \end{aligned}$$
However, since \(\pi _*\mathcal {O}(D)= \mathcal {O}\oplus \omega ^\vee \), we have
$$\begin{aligned} \pi _*(K(D))\simeq {\text {Ker }}\left( (\mathcal {O}\oplus \omega ^\vee )\otimes (\mathcal {O}\oplus \omega ^\vee )\rightarrow \mathcal {O}\oplus \omega ^\vee \oplus (\omega ^\vee )^2\right) \simeq \omega ^\vee . \end{aligned}$$
Therefore \(K\simeq \pi ^*\pi _*(K(D))\otimes \mathcal {O}(-D)\simeq \pi ^*\omega ^\vee (-D)\), and we may conclude. \(\square \)
Remark 7.6
For later purposes, let us fix an isomorphism \(\pi _*(K(D))\simeq \omega ^\vee \), so that the inclusion \(\pi _*(K(D))\subset \pi _*\mathcal {O}(D)\otimes \pi _*\mathcal {O}(D)\) is identified with the composition
$$\begin{aligned}\omega ^\vee \xrightarrow {(1,-1)}\left( \omega ^\vee \otimes \mathcal {O}\right) \oplus \left( \mathcal {O}\otimes \omega ^\vee \right) \subset \left( \mathcal {O}\oplus \omega ^\vee \right) \otimes \left( \mathcal {O}\oplus \omega ^\vee \right) . \end{aligned}$$
Let us return to the equivalence (29). Fix \(d>0\), and a locally free sheaf \(\mathcal F\in {\text {Coh }}C\) of rank n. Consider subfunctors \(\underline{\mathcal Coh}_d^{\mathcal F}S\subset \underline{\mathcal D^b({\text {Coh }}S)}\), \(\underline{\mathcal A\text {-}mod}_d^{\mathcal F}\subset \underline{\mathcal A\text {-}mod_{\mathcal D}}\), defined as follows:
Proposition 7.7
For any \(d>0\), the equivalence (29) induces a natural transformation
$$\begin{aligned} \Theta _d'{:}\,\underline{\mathcal Coh}_d^{\mathcal F}S\rightarrow \underline{\mathcal A\text {-}mod}_d^{\mathcal F}. \end{aligned}$$
Proof
We need to check that for any T, we have \(\Theta \left( \underline{\mathcal Coh}_d^{\mathcal F}S(T) \right) \subset \underline{\mathcal A\text {-}mod}_d^{\mathcal F}(T)\). Let \(E\in \underline{\mathcal Coh}_d^{\mathcal F}S(T)\). Then by definition \(\Theta (E)=(R\pi _*E(-D),R\pi _*E,\varphi _0,\varphi _1)\), where \(\varphi _0{:}\,R\pi _*E(-D)\rightarrow R\pi _*E\) is obtained by applying \(R\pi _*\) to the first map in the short exact sequence
$$\begin{aligned} 0\rightarrow E(-D)\rightarrow E\rightarrow E|_{D\times T}\rightarrow 0. \end{aligned}$$
In particular, \(Cone(\varphi _0)\simeq R\pi _*(E|_{D\times T})\simeq \mathcal F\boxtimes \mathcal {O}_T\).
Pick a point \(t\in T\). We have \(\pi _*E(-D)_t=0\) by Lemma 7.4, so that \(R\pi _*E(-D)_t=R^1\pi _*E(-D)_t[-1]\). Moreover, the formulas (30) applied to \(E(-D)_t\) show that \({\text {rk }}(R\pi _*E(-D)_t)=0\) and \(c_1 (R\pi _*E(-D)_t)=-d\). Therefore, \(R^1\pi _*E(-D)_t\) is a torsion sheaf of degree d.
Finally, let us prove flatness. Let \(f{:}\,T\times S\rightarrow T\), \(p{:}\,T\times C\rightarrow T\) be the natural projections. Using the second part of Lemma 7.4, a proof analogous to [29, Corollary 4.2.12] shows that \(R^1f_*E(-D)\) is a locally free sheaf. Let \(\mathcal L\) be an ample line bundle on C, and \(k\in \mathbb {N}\). Since \(R^1\pi _*E(-D)_t\) is a torsion sheaf for any \(t\in T\), it is isomorphic to \(\mathcal L^k\otimes R^1\pi _*E(-D)\) in the neighborhood of t. In particular, the fact that \(p_*(R^1\pi _*E(-D))\simeq R^1f_*E(-D)\) is locally free implies that \(p_*(\mathcal L^k\otimes R^1\pi _*E(-D))\) is locally free for any k. We conclude that \(R^1\pi _*E(-D)\) is flat over T by [21, Proposition 2.1.2]. \(\square \)
Consider rigidified functors \(\underline{\mathcal Coh}_d^{\leftarrow \mathcal F}S\), \(\underline{\mathcal D^b({\text {Coh }}S)}^{\leftarrow \mathcal F}\), where we fix the additional data of an isomorphism \(\Psi {:}\,E|_D\xrightarrow {\sim } \mathcal F\). We will refer to elements of \((\underline{\mathcal Coh}_d^{\leftarrow \mathcal F}S)(\mathbb {k})\) as \(\mathcal F\)-framed sheaves.
Lemma 7.8
Any \(\mathcal F\)-framed sheaf is locally free in a neighborhood of D.
Proof
Recall that for any torsion-free sheaf E, its double dual \(E^{\vee \vee }\) is a vector bundle. Let \((E,\Psi )\) be an \(\mathcal F\)-framed sheaf, and consider the quotient \(U=E^{\vee \vee }/E\). It is a sheaf with zero-dimensional support. If the intersection \(D\cap {\text {supp }}U\) is non-empty, \(E|_D\) is a proper subsheaf of \(E^{\vee \vee }|_D\). However,
$$\begin{aligned} \deg E|_D=\deg \mathcal F=c_1(E)D=c_1(E^{\vee \vee })D=\deg E^{\vee \vee }|_D, \end{aligned}$$
and \({\text {rk }}E|_D={\text {rk }}E^{\vee \vee }|_D\), so that \(E|_D=E^{\vee \vee }|_D\). Therefore the support of U is disjoint from D, and we have an isomorphism \(E\simeq E^{\vee \vee }\) in a neighborhood of D. \(\square \)
Let us recall a closely related notion of stable pairs. We specialize the definition in [5] to the case when polarization of S is given by the divisor \(H=D+Nf\), and \(N>2g-2\). Recall that for any locally free sheaf \(\mathcal E\) on C, its slope is defined as \(\mu (\mathcal E)=\deg \mathcal E/{\text {rk }}\mathcal E\).
Definition 7.9
Let E be a torsion-free sheaf on S satisfying \({\text {ch }}E=(n,\deg \mathcal F\cdot f,-d)\), and \(\Psi {:}\,E|_D\xrightarrow {\sim }\mathcal F\) an isomorphism. Fix \(N>2g-2\), and \(\delta >0\). A pair \((E,\Psi )\) is said to be \((N,\delta )\)-stable, if for any subsheaf \(E'\subset E\) with \(0<{\text {rk }}E'<n\) the following inequality holds:
$$\begin{aligned} \frac{c_1(E')H}{{\text {rk }}E'}< {\left\{ \begin{array}{ll} \mu (\mathcal F)-\delta /n &{}\quad \hbox {if}\,E'\subset E(-D), \\ \mu (\mathcal F)+\delta /{\text {rk }}E'-\delta /n &{}\quad \hbox {otherwise}. \end{array}\right. } \end{aligned}$$
It is known that the moduli of \((N,\delta )\)-stable pairs is represented by a quasi-projective variety, see [5, Theorem 2.3].
Proposition 7.10
There exist \(N,\delta \) big enough, such that every \(\mathcal F\)-framed sheaf \((E,\Psi )\) is \((N,\delta )\)-stable. In particular, the functor \(\underline{\mathcal Coh}_d^{\leftarrow \mathcal F}S\) is represented by a quasi-projective variety \({\mathscr {B}}(d,\mathcal F)\).
Proof
The \((N,\delta )\)-stability condition is vacuous for sheaves of rank 1. Therefore, we will assume that \(n\ge 2\). The existence of Harder-Narasimhan filtration [29, Chapter 5] implies that for a locally free sheaf \(\mathcal F\) on C, there exists a constant \(\mu _{max}(\mathcal F)\), such that \(\mu (\mathcal F')<\mu _{max}(\mathcal F)\) for all \(\mathcal F'\subset \mathcal F\). From now on, we will assume that \(\delta >(\mu _{max}(\mathcal F)-\mu (\mathcal F))n^2\), and \(N>2g-2+\delta \).
Let \(E'\subset E\) be a subsheaf of rank \(n'\) with \(0<n'<n\). Since \(E_c\simeq \mathcal {O}_{\mathbb {P}^1}^n\) for a generic \(c\in C\), there exist integers \(0\le k_{1}\le \cdots \le k_{n'}\) such that generically \(E'_c\simeq \bigoplus _{i=1}^{n'}\mathcal {O}_{\mathbb {P}^1}(-k_i)\).
Assume first that \(E'\not \subset E(-D)\). Consider the saturation \(\overline{E'}\) of \(E'\) inside E. It has the same rank as \(E'\), and \(c_{1,D}(\overline{E'})\le 0\). Moreover, since E is a vector bundle in the neighborhood of D by Lemma 7.8, \(\overline{E'}\) is its subbundle in the same neighborhood. As a consequence, we have \(\overline{E'}|_D\subset E|_D\), and \(c_{1}(\overline{E'})D=\deg (\overline{E'}|_D)\). Putting this together, we obtain
$$\begin{aligned} \begin{aligned} \frac{c_1(E')H}{r'}&\le \frac{c_1(\overline{E'})H}{r'}=\mu \left( \overline{E'}|_D\right) +\frac{Nc_{1,D}(\overline{E'})}{r'}\le \mu _{max}(\mathcal F)<\mu (\mathcal F)+\delta /n^2\\&<\mu (\mathcal F)+\delta /r'-\delta /n, \end{aligned} \end{aligned}$$
(31)
which is the desired estimate.
Now, suppose \(E'\subset E(-D)\). In this case \(k_1>0\), and \(E'\) is not contained in \(E(-(k_1+1)D)\). Let k be the maximal positive integer such that \(E'\subset E(-kD)\); we have \(k\le k_1\). In particular, \(E'(kD)\) is naturally a subsheaf of E, which is not contained in \(E(-D)\). Moreover, for a generic point \(c\in C\), we have \(E'(kD)_c\simeq \bigoplus _{i=1}^{n'}\mathcal {O}_{\mathbb {P}^1}(-{\widetilde{k}}_i)\), where \({\widetilde{k}}_i=k_i-k\ge 0\) for all i. Therefore, the inequality (31) holds for \(E'(kD)\) by previous considerations. We have
$$\begin{aligned} \frac{c_1(E')H}{r'}&=\frac{c_1(E'(kD))H}{r'}-kD\cdot H =\frac{c_1(E'(kD))H}{r'}+k(2g-2-N)\\&<\mu (\mathcal F)+\delta /n^2+k(2g-2-N)<\mu (\mathcal F)+\delta /n^2-k\delta <\mu (\mathcal F)-\delta /n. \end{aligned}$$
We can thus conclude that for our choice of \(N,\delta \) every \(\mathcal F\)-framed sheaf is \((N,\delta )\)-stable. \(\square \)
Remark 7.11
The divisor \(D\subset S\) is not nef when \(g(C)>1\), so that [5, Theorem 3.1] could not be invoked directly. Moreover, one can check that if we omit framing, f-semistable sheaves can possess automorphisms.
Analogously to \(\underline{\mathcal Coh}_d^{\leftarrow \mathcal F}S\), consider the rigidified functor \(\underline{\mathcal A\text {-}mod}_d^{\leftarrow \mathcal F}\), given by fixing a distinguished triangle \(\Delta = \left( \mathcal V_1[1]\xrightarrow {\varphi _0} \mathcal V_2^\bullet \xrightarrow {\psi } \mathcal F\xrightarrow {\varphi '_0}\right) \). Then \(\Theta _d'\) extends to a natural transformation \(\Theta ^\leftarrow =\Theta ^\leftarrow _d{:}\, \underline{\mathcal Coh}_d^{\leftarrow \mathcal F}S\rightarrow \underline{\mathcal A\text {-}mod}_d^{\leftarrow \mathcal F}\).
Let us establish relation between \(\underline{\mathcal A\text {-}mod}_d^{\leftarrow \mathcal F}\) and the stack of Higgs triples.
Lemma 7.12
Let \(\mathcal E,\mathcal F\in {\text {Coh }}C\), \(\varphi \in {\text {Hom }}(\mathcal F, \mathcal E)\), and \(\mathcal C_1, \mathcal C_2\) two cones of \(\varphi \). Then there exists the unique map \(f\in \mathbb {H}\hbox {om}(\mathcal C_1,\mathcal C_2)\) making the following diagram commute:
where i, j are the natural maps.
Proof
The existence of map f is assured by axioms of triangulated category. Let \(f_1\), \(f_2\) be two such maps, and consider their difference \(g=f_1-f_2{:}\,\mathcal C_1\rightarrow \mathcal C_2\). By definition, we have \(g\circ i=0\) and \(j\circ g=0\). Therefore, g lies in the image of composition
$$\begin{aligned} \mathbb {H}\hbox {om}(\mathcal F[1],\mathcal E)\rightarrow \mathbb {H}\hbox {om}(\mathcal C_1,\mathcal E)\rightarrow \mathbb {H}\hbox {om}(\mathcal C_1,\mathcal C_2). \end{aligned}$$
Since both \(\mathcal E\) and \(\mathcal F\) lie in the heart of \(\mathcal D^b({\text {Coh }}C)\), we have \(\mathbb {H}\hbox {om}(\mathcal F[1],\mathcal E)=0\). Thus \(g=0\), and the unicity of f follows. \(\square \)
Thanks to the lemma above, we can define a natural transformation \(\tau {:}\,\underline{\mathcal A\text {-}mod}_d^{\leftarrow \mathcal F}\rightarrow T^*\underline{\mathcal Coh}^{\leftarrow \mathcal F}_{0,d}\), which to each element \((\mathcal V_1[-1],\mathcal V_2^\bullet ,\varphi _0,\varphi _1,\Delta )\in \underline{\mathcal A\text {-}mod}_d^{\leftarrow \mathcal F}(T)\) associates the triple
$$\begin{aligned} (\mathcal V_1, \varphi '_0, f\circ \varphi _1), \end{aligned}$$
where \(f{:}\,\mathcal V_2^\bullet \xrightarrow {\sim } (\mathcal F\xrightarrow {\varphi _0} \mathcal V_1)\) is the unique isomorphism given by Lemma 7.12.
Proposition 7.13
The functor \(\tau \) is a natural equivalence.
Proof
Let us consider a natural transformation
$$\begin{aligned} \upsilon {:}\, T^*\underline{\mathcal Coh}^{\leftarrow \mathcal F}_{0,d}\rightarrow \underline{\mathcal A\text {-}mod}_{d}^{\leftarrow \mathcal F}, \end{aligned}$$
defined on T-points by the following formula:
$$\begin{aligned} \upsilon (\mathcal E, \alpha ,\theta )=\left( \mathcal E[-1],\mathcal F\xrightarrow {\alpha }\mathcal E,\iota ,\theta [-1],\Delta \right) . \end{aligned}$$
Here \(\iota \) is the natural map \(\mathcal E[-1]\rightarrow (\mathcal F\rightarrow \mathcal E)\), and \(\Delta \) is obtained from the mapping cone of \(\alpha \):
$$\begin{aligned} \Delta = \left( \mathcal E[-1]\xrightarrow {\iota } (\mathcal F\rightarrow \mathcal E) \xrightarrow {\psi } \mathcal F\xrightarrow {\alpha }\right) . \end{aligned}$$
It is clear that \(\tau \circ \upsilon \) is the identity functor. On the other hand, the composition \(\upsilon \circ \tau \) sends \((\mathcal V_1,\mathcal V_2^\bullet ,\varphi _0,\varphi _1,\Delta )\) to \((\mathcal V_1,\mathcal F\xrightarrow {\varphi '_0}\mathcal V_1,f\circ \varphi _0,f\circ \varphi _1,\Delta ')\), where \(\Delta '\) is the triangle
$$\begin{aligned} \Delta ' = \left( \mathcal V_1[-1]\xrightarrow {\iota } (\mathcal F\rightarrow \mathcal V_1) \xrightarrow {\psi } \mathcal F\xrightarrow {\varphi _0'}\right) . \end{aligned}$$
The map f induces a natural equivalence \(\upsilon \circ \tau \simeq {\text {Id }}_{\underline{\mathcal A\text {-}mod}_{d}^{\leftarrow \mathcal F}}\), so that \(\tau \) and \(\upsilon \) are mutually inverse equivalences. \(\square \)
Theorem 7.14
The composition \(\overleftarrow{\Theta } =\tau \circ \Theta ^\leftarrow \) factors through the stack of stable Higgs triples, and induces an equivalence
$$\begin{aligned} \overleftarrow{\Theta }{:}\,\underline{\mathcal Coh}_d^{\leftarrow \mathcal F}S\xrightarrow {\sim }\left( T^*\underline{\mathcal Coh}^{\leftarrow \mathcal F}_{0,d} \right) ^{st}. \end{aligned}$$
Proof
Since the functor \(\overleftarrow{\Theta }\) is fully faithful, it is enough to compute its image on T-points of \(\underline{\mathcal Coh}_d^{\leftarrow \mathcal F}S\) for every T. Let \((\mathcal E,\alpha ,\theta )\) be a Higgs triple, and consider the morphism
$$\begin{aligned} \xi =\varepsilon \otimes 1+\pi ^*(\iota ,\theta \otimes \omega ^\vee )&\in \mathbb {H}\hbox {om}\left( \pi ^*\pi _*\mathcal {O}(D)\otimes \pi ^*\mathcal E, \mathcal {O}(D)\otimes \pi ^*\mathcal E\oplus \pi ^*\alpha [1]\right) , \end{aligned}$$
where \(\varepsilon {:}\,\pi ^*\pi _*\mathcal {O}(D)\rightarrow \mathcal {O}(D)\) is the counit map from Lemma 7.5, and we use the identification \(\pi _*\mathcal {O}(D)\simeq \mathcal {O}\oplus \omega ^\vee \). After restricting to D, we get
$$\begin{aligned} \xi |_D=\begin{pmatrix} \iota &{}\quad \theta \otimes \omega ^\vee \\ 0 &{}\quad 1 \end{pmatrix}{:}\,\mathcal E\oplus \mathcal E\otimes \omega ^\vee \rightarrow \alpha [1]\oplus \mathcal E\otimes \omega ^\vee . \end{aligned}$$
The map \(\xi |_D\) can be naturally completed to a distinguished triangle
$$\begin{aligned} \mathcal F\xrightarrow {(\alpha ,0)}\mathcal E\oplus \mathcal E\otimes \omega ^\vee \xrightarrow {\xi |_D} \alpha [1]\oplus \mathcal E\otimes \omega ^\vee \xrightarrow {+1}. \end{aligned}$$
Thus, we obtain an identification \(\Psi {:}\,Cone(\xi |_D)\xrightarrow {\sim } \mathcal F\).
Note that
$$\begin{aligned} \overleftarrow{\Theta }(E)=\tau \circ F'(R\pi _*\mathcal Hom(\mathcal T,E)) \end{aligned}$$
up to remembering the framing. Combining the inverses of each functor in the composition provided by Propositions 7.1, 7.13, and Lemma 6.5, we obtain the following left inverse of \(\Theta ^\leftarrow \) on the set of isomorphism classes of T-points:
$$\begin{aligned} G_T{:}\,T^*\underline{\mathcal Coh}^{\leftarrow \mathcal F}_{0,d}(T) \rightarrow \underline{\mathcal D^b({\text {Coh }}S)}^{\leftarrow \mathcal F}(T),\qquad (\mathcal E,\alpha ,\theta ) \mapsto (Cone(\kappa \circ \xi )[-1],\Psi ), \end{aligned}$$
where \(\kappa {:}\,\pi ^*\mathcal E(D)\oplus \pi ^*\alpha [1]\rightarrow \pi ^*\mathcal E(D)\oplus \pi ^*\alpha [1]\) multiplies first summand by 1, and the second one by \(-1\). In what follows, we are only interested in the set-theoretic image of \(G_T\). As such, we will not concern ourselves with functoriality, and will liberally make use of (non-unique) cones of various maps.
Lemma 7.15
As a complex, \(\xi \) is quasi-isomorphic to \(\mathcal {O}(-D)\otimes (\pi ^*(\mathcal E\otimes \omega ^\vee )\xrightarrow {\xi '} \mathcal {O}(D)\otimes \pi ^*\alpha [1])\), where \(\xi '\) is obtained by adjunction from \((-\theta \otimes \omega ^\vee ,\iota \otimes \omega ^\vee )\in \mathbb {H}\hbox {om}(\mathcal E\otimes \omega ^\vee ,\pi _*\mathcal {O}(D)\otimes \alpha [1])\).
Proof
Let \(p_1,p_2\) be the projection maps from \(\pi ^*\mathcal E(D)\oplus \pi ^*\alpha [1]\) to the first and the second summand respectively. Note that \(p_1\circ \xi =\varepsilon \otimes 1\). Applying octahedral axiom to these three maps, we obtain a distinguished triangle, denoted by dashed arrows below:
Here, j is defined by Lemma 7.5, and \(i_2\) is the natural inclusion of a summand. We see that \(\xi \) is quasi-isomorphic to \(\xi ''\), so it remains to compute the map \(\xi ''\). Since j is injective and the diagram commutes, we have \(\xi ''=p_2\circ \xi \circ j=\pi ^*(\iota ,\theta \otimes \omega ^\vee )\circ j\). After tensoring with \(\mathcal {O}(D)\) and applying \(\pi _*\), we obtain the composition
$$\begin{aligned} \mathcal E\otimes \omega ^\vee \hookrightarrow (\mathcal {O}\oplus \omega ^\vee )\otimes (\mathcal {O}\oplus \omega ^\vee )\otimes \mathcal E\xrightarrow {1\otimes (\iota ,\theta \otimes \omega ^\vee )}(\mathcal {O}\oplus \omega ^\vee )\otimes \alpha [1]. \end{aligned}$$
The map on the left is induced by the diagonal embedding \(\omega ^\vee \xrightarrow {(1,-1)} \omega ^\vee \otimes \omega ^\vee \) as in Remark 7.6. Thus the composition is precisely \((-\theta \otimes \omega ^\vee ,\iota \otimes \omega ^\vee )\), and we may conclude. \(\square \)
Let us consider the map \(G_\mathbb {k}\) between \(\mathbb {k}\)-points. Recall (see Sect. 5) that as a complex, \(\alpha \) is quasi-isomorphic to \(K\oplus J[-1]\), where \(K={\text {Ker }}\alpha \), \(J={\text {Coker }}\alpha \). Thus, we can express \(\xi '\) as a sum:
$$\begin{aligned} \xi '=\xi '_e+\xi '_h,\qquad \xi '_e{:}\,\pi ^*(\mathcal E\otimes \omega ^\vee )\rightarrow \pi ^*K(D)[1],\qquad \xi '_h{:}\,\pi ^*(\mathcal E\otimes \omega ^\vee )\rightarrow \pi ^*J(D). \end{aligned}$$
Let \(M\in {\text {Ext }}^1(\pi ^*(\mathcal E\otimes \omega ^\vee ),\pi ^*K(D))\) be the extension given by \(\xi '_e\). Then the two-step complex given by \(\xi '\) is quasi-isomorphic to \(M\rightarrow \pi ^*J(D)\), with arrow defined as the composition of \(\xi '_h\) with the projection \(M\twoheadrightarrow \pi ^*(\mathcal E\otimes \omega ^\vee )\). Consequently, the cone of \(\xi '\) has length 1 if and only if \(\xi '_h\) is surjective.
Lemma 7.16
The map \(\xi '_h\) is surjective if and only if the triple \((\mathcal E,\alpha ,\theta )\) is stable.
Proof
By abuse of notation, we will write \(\xi =\xi '_h\otimes \omega \) throughout the proof, and study surjectivity of \(\xi \). Thanks to Lemma 7.15, \(\xi \) is adjoint to
$$\begin{aligned} (\theta ,-\iota ){:}\,\mathcal E\rightarrow J\otimes (\omega \oplus \mathcal {O}), \end{aligned}$$
where \(\theta =\theta _h\) as in Section 5, and \(\iota =\iota _h{:}\,\mathcal E\twoheadrightarrow J\) is the natural projection. Let \(c\in C\), and \(s=\pi ^{-1}(c)=\mathbb {P}(\omega _c\oplus \mathcal {O}_c)\). If we choose an identification \(\omega _c\simeq \mathcal {O}_c\), the stalk \(\xi _s\) at the point s is given by a linear combination \(a\theta _c+b\iota _c\) for some \(a,b\in \mathbb {k}\). In particular, \(\xi \) is surjective if and only if it is surjective at each point \(s\in S\), that is for every \(c\in C\) and \([a{:}\,b]\in \mathbb {P}^1\) the map \(a\theta _c+b\iota _c\) is surjective.
Suppose that \(\xi \) is not surjective. Then there exists a point \(s\in \pi ^{-1}(c)\), \(c\in C\) where surjectivity fails. This means that \(J'_c:={\text {Im }}(a\theta _c+b\iota _c)\) is a proper subsheaf of \(J_c\) for some \(a\ne 0\), b. Denote \(\mathcal E'_c=\iota ^{-1}(J'_c)\); then \(\theta _c(\mathcal E'_c)\subset J'_c\). Further, let
$$\begin{aligned} \mathcal E'=\mathcal E'_c\oplus \bigoplus _{p\in C{\setminus }\{c\}}\mathcal E_p,\quad J'=J'_c\oplus \bigoplus _{p\in C{\setminus }\{c\}}J_p. \end{aligned}$$
Then \({\text {Im }}\alpha \subset \mathcal E'\) and \(\theta (\mathcal E')\subset J'\), which precludes the triple \((\mathcal E,\alpha ,\theta )\) from being stable.
Conversely, suppose that \((\mathcal E,\alpha ,\theta )\) is destabilized by a subsheaf \(\mathcal E'\subset \mathcal E\). Denote \(J'=\iota (\mathcal E')\). Let us choose a point \(c\in C\), such that \(\mathcal E'_c\subset \mathcal E_c\) is a proper subsheaf. By assumption \(\xi (\mathcal E'_c)\subset J'_c\) and \(\mathcal E/\mathcal E'\simeq J/J'\). Suppose \(\xi \) is surjective. Then for any \(s\in \pi ^{-1}(c)\) the stalk \(\xi _s\) induces an automorphism of \(\mathcal E_c/\mathcal E'_c\). In particular, the map \(a{\text {Id }}+b\theta _c\) is an automorphism of \(\mathcal E_c/\mathcal E'_c\) for each \([a{:}\,b]\in \mathbb {P}^1\). However, \(\mathcal E\) is a torsion sheaf, therefore \(\mathcal E_c/\mathcal E'_c\) is finite-dimensional as a \(\mathbb {k}\)-module. Because of this, \(\theta \) must possess an eigenvalue \(\lambda \), so that \(\theta -\lambda {\text {Id }}\) cannot be invertible. Thus \(\xi \) is not surjective. \(\square \)
Lemma 7.16 shows that any family of Higgs triples which contains a non-stable one is mapped outside of \(\underline{\mathcal Coh}_d^{\leftarrow \mathcal F}S\) by \(G_T\). This proves that the essential image of \(\overleftarrow{\Theta }\) is contained in \(\left( T^*\underline{\mathcal Coh}^{\leftarrow \mathcal F}_{0,d} \right) ^{st}\). We now need to show that every flat T-family \((\mathcal E_T,\alpha _T,\theta _T)\) of stable Higgs triples lies in the image of \(\overleftarrow{\Theta }\), or equivalently its image \(G_T(\mathcal E_T,\alpha _T,\theta _T)\) lies in \(\underline{\mathcal Coh}_d^{\leftarrow \mathcal F}S(T)\). By Lemma 7.16, it is a coherent sheaf \(E_T\) on \(S\times T\), equipped with an isomorphism \(\Psi {:}\,E_T|_{D\times T}\xrightarrow {\sim }\mathcal F\boxtimes \mathcal {O}_T\). Moreover, by construction \(E_T\) is a subsheaf of \(M_T\), with latter being obtained as an extension of a torsion-free sheaf \(\pi ^*K_T(D)\) by \(\pi ^*\mathcal E_T\). As a consequence, the torsion \({\text {Tor }}E_T\) is contained in the support of \(\pi ^*\mathcal E_T\). However, since \(\mathcal F\) is locally free, the existence of \(\Psi \) implies that the support of \({\text {Tor }}E_T\) must be disjoint from \(D\times T\). Since the support of every subsheaf of \(\pi ^*\mathcal E\) intersects \(D\times T\), we conclude that E is torsion-free.
Pick a point \(t\in T\). Outside of the support of \(\pi ^*\mathcal E\), the complex \(\xi [-1]\) is quasi-isomorphic to \(\pi ^*\mathcal F\). By Lemma 7.3, it implies that \(E_t\) is f-stable.
Let us compute the Chern character of \(E_t\):
$$\begin{aligned} {\text {ch }}(E_t)&={\text {ch }}(\mathcal {O}(-D)\otimes \pi ^*(\mathcal E_t\otimes \omega ^\vee ))-{\text {ch }}(\pi ^*\alpha _t[1])\\&=(1,-D,1-g)(0,df,0)-(0,df,0)-(n,\deg F\cdot f,0) =(n,\deg F\cdot f,-d). \end{aligned}$$
Thus \(c_1(E_t)=\deg F\cdot f\), and \(c_2(E_t)=c_1(E_t)^2/2-{\text {ch }}_2(E_t)=d\).
It remains to show that \(E_T\) is T-flat. For this, we will express \(E_T\) in a different fashion. In what follows, we will drop the subscript T, implicitly assuming that all objects live in families over T.
Denote by \(ev{:}\,H^0(\mathcal E)\otimes \mathcal {O}\twoheadrightarrow \mathcal E\) the natural evaluation map. Let \(\widetilde{\mathcal K}\) be the kernel of
$$\begin{aligned} (\alpha ,ev){:}\,\mathcal F\oplus \left( H^0(\mathcal E)\otimes \mathcal {O}\right) \twoheadrightarrow \mathcal E. \end{aligned}$$
The octahedral axiom applied to the composition \(\mathcal F\xrightarrow {({\text {id }},0)}\mathcal F\oplus H^0(\mathcal E)\otimes \mathcal {O}\xrightarrow {(\alpha ,ev)} \mathcal E\) produces a distinguished triangle
$$\begin{aligned} \pi ^*\alpha \rightarrow \pi ^*\widetilde{\mathcal K}\rightarrow H^0(\mathcal E)\otimes \mathcal {O}_S\xrightarrow {+1}. \end{aligned}$$
Next, consider the composition \(H^0(\mathcal E)\otimes \mathcal {O}_S[-1]\rightarrow \pi ^*\alpha \rightarrow E\), where the first map is defined by the triangle above, and the second map comes from the quasi-isomorphism \(E\simeq \xi '\otimes (-D)\). One more application of the octahedral axiom gives rise to the following diagram:
Here, \({\widetilde{E}}\) is defined as a cone of the composition above. Note that since both E and \(H^0(\mathcal E)\otimes \mathcal {O}_S\) are sheaves (as opposed to complexes of sheaves), \({\widetilde{E}}\) is also a sheaf.
Recall that if N is a T-flat sheaf, and
$$\begin{aligned} 0\rightarrow M_1\rightarrow M_2\rightarrow N\rightarrow 0 \end{aligned}$$
is a short exact sequence, then \(M_1\) is T-flat if and only if \(M_2\) is. As a consequence of this, \(\pi ^*\widetilde{\mathcal K}\) is T-flat as the kernel of \(\pi ^*(\alpha ,ev)\); the middle row of diagram (32) shows \({\widetilde{E}}\) is T-flat; and finally, the middle column of (32) shows that E is T-flat as well. \(\square \)
Proof of Theorem 5.8
Representability follows from Theorem 7.14 together with Proposition 7.10. For smoothness, recall [5, Theorem 4.3] that \({\mathscr {B}}(d,\mathcal F)\) is smooth at a point \((E,\Psi )\) if the kernel of the trace map
$$\begin{aligned} {\text {Ext }}^2(E,E(-D))\rightarrow H^2(S,\mathcal {O}(-D)) \end{aligned}$$
vanishes. Since \({\text {Ext }}^2(E,E(-D))\simeq {\text {Hom }}(E,E(-D))^*\) by Serre duality, and \(H^2(S,\mathcal {O}(-D))\simeq {\text {Hom }}(\mathcal {O},\mathcal {O}(-D))^*=0\), it suffices to show that for any f-stable sheaf E there exist no non-zero maps from E to \(E(-D)\). Let \(\varphi {:}\,E\rightarrow E(-D)\) be such a map. By definition of f-stability, \(E|_{\pi ^{-1}(c)}\simeq \mathcal {O}_{\mathbb {P}^1}^n\) for a generic point \(c\in C\). Since \(\mathcal {O}_{\mathbb {P}^1}(-1)\) has no global sections, this implies that the image of \(\varphi \) must be a torsion sheaf. However, \(E(-D)\) is a torsion-free sheaf, so we may conclude.
For the second claim, let \(E_1\) be a locally free sheaf of rank 1 on S, such that \(c_1(E_1)=0\). By the seesaw principle, \(E_1\simeq \pi ^*\pi _*E_1\). In particular, if \(E_1|_D\simeq \mathcal {O}_C\), then \(E_1\simeq \mathcal {O}_S\). As a consequence, we have \(E^{\vee \vee }\simeq \mathcal {O}_S\) for any \((E,\Psi )\in {\mathscr {B}}(d,1)\), and fixing \(\Psi \) makes this isomorphism canonical. Therefore, the map
$$\begin{aligned} E\mapsto (E|_{T^*C})\subset E^{\vee \vee }|_{T^*C}=\mathcal {O}_{T^*C} \end{aligned}$$
establishes the desired isomorphism \({\mathscr {B}}(d,1)\simeq {\text {Hilb }}_dT^*C\). \(\square \)
In view of Theorem 5.8, it is instructive to compare our results with recent works of Neguţ [38, 39]. For any smooth projective surface S and an ample divisor H, he considers the moduli space \(\mathcal M\) of H-stable sheaves on S with varying second Chern class, and for every \(n\in \mathbb {Z}\) defines an operator \(e_n{:}\,K(\mathcal M)\rightarrow K(\mathcal M\times S)\) by Hecke correspondences. These operators generate a subalgebra \(\mathcal A\) inside \(\bigoplus _{k>0}{\text {Hom }}(K(\mathcal M)\rightarrow K(\mathcal M\times S^k))\), which can be then projected to a shuffle algebra \(\mathcal V_{sm}\). The content of Conjecture 3.20 in [39] is that this projection is supposed to be an isomorphism. This conjecture is proved under rather restrictive assumptions; for instance, it is required that \(K(S\times S)\simeq K(S)\otimes K(S)\).
Let us now take \(S=T^*C\) together with a scaling action of \(T\simeq \mathbb {G}_m\), and replace usual K-groups with their T-equivariant counterpart. In this case, the algebra \(\mathcal V_{sm}\) can be identified with the subalgebra of \(K\mathbf{Sh}^{norm}_C\), generated by \(K(\mathrm {B}\mathbb {G}_m)\subset K\mathbf{Sh}^{norm}_C[1]\simeq K^T(C\times \mathrm {B}\mathbb {G}_m)\). If we further replace K-groups by Borel–Moore homology, then by Corollary 4.5 homological version of \(\mathcal V_{sm}\) is realized as a subalgebra of \(H\mathbf{Ha}_C^{0,T}\). Therefore, one can regard results of Sect. 5 as a “homological non-compact” version of Neguţ’s conjecture for \(S=T^*C\), \(c_{1,D}=0\), and stability condition given by f. Another modest gain of our approach is that while \(\mathcal A\) is given by operators on K-groups \(K(\mathcal M)\), the definition of \(A\mathbf{Ha}_C^{0,T}\) is independent from its natural representations, which allows to study this algebra without invoking torsion-free sheaves on \(T^* C\).
In general, one expects that the moduli of framed sheaves on \(\mathbb {P}_C(\omega \oplus \mathcal {O})\) with non-trivial first Chern class can be recovered from the moduli of stable Higgs triples of positive rank. Nevertheless, as stability condition for triples varies, Lemma 7.16 seems to suggest that the objects on S which correspond to stable Higgs sheaves do not have to lie in the usual heart of \(\mathcal D^b({\text {Coh }}S)\). These questions will be investigated in future work.