On the Bogomolov–Gieseker inequality for tame Deligne–Mumford surfaces

Abstract

We generalize the Bogomolov–Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne–Mumford surfaces. We work over positive characteristic $p>0$ and generalize Langer’s method to smooth Deligne–Mumford stacks. As applications we obtain the Bogomolov inequality for semistable coherent sheaves on a Deligne–Mumford surface in characteristic zero, and the Bogomolov inequality for semistable sheaves on a root stack over a smooth surface which is equivalent to the Bogomolov inequality for the rational parabolic sheaves on a smooth surface S. In a joint appendix with Hao Sun, we generalize the Bogomolov inequality formula to Simpson Higgs sheaves on tame Deligne–Mumford stacks.

Appendices

Appendix A: Higher dimension case

In the appendix we generalize Langer’s argument to higher dimension case. For simplicity of the calculation of modified slopes, we restrict to a special case of smooth Deligne–Mumford stacks $\mathcal {X}=[Z/G]$ which is a quotient stack such that the action of G is diagonalizable. We still let the generating sheaf $\**$ on $\mathcal {X}$ satisfying condition Sect. 3.1. Still let $d=\dim (\mathcal {X})$ be the dimension of $\mathcal {X}.$

We state several theorems generalizing Langer [15, §3].

Theorem A.1

Let $D_1$ be a very ample divisor on X and ${{\mathcal {D}}}_1:=\pi ^{-1}(D_1)$ and ${{\mathcal {D}}}=\pi ^{-1}(D)$ for D a general element $D\in |D_1|$. If the restriction of a coherent sheaf E on $\mathcal {X}$ to ${{\mathcal {D}}}$ is not modified slope semistable with respect to $H|_{D}$ and $\** |_{D},$ then let $\mu _{\** ,i}, r_i$ denote the modified slopes and ranks respectively in the Harder–Narasimhan filtration of $E|_{D},$ we have

$$\begin{aligned}{} & {} \sum _{i<j}r_i r_j(\mu _{\** ,i}-\mu _{\** ,j})^2\leqslant H^{d}\Delta (E)\nonumber \\{} & {} \quad +2\mathop {\text {rk}}(E)^2(L_{\** ,\max }(E)-\mu _{\** }(E)) (\mu _{\** }(E)-L_{\** ,\min }(E)). \end{aligned}$$

(A.1)

(See 2.3 for definitions related to the above theorem.)

Theorem A.2

If a torsion free sheaf E on $\mathcal {X}$ is strongly modified slope semistable,  we have

$$\begin{aligned} \Delta (E)\cdot H^{d-2}\geqslant 0. \end{aligned}$$

Theorem A.3

If a torsion free sheaf E on $\mathcal {X}$ is just modified slope semistable,  then we have

$$\begin{aligned} H^{d}\cdot \Delta (E)\cdot H^{d-2}+\mathop {\text {rk}}(\** )^2\beta _{\mathop {\text {rk}}(E)}\geqslant 0. \end{aligned}$$

Before stating the last theorem, we introduce some notations. First for torsion free sheaves $G^{\prime }, G$ on $\mathcal {X},$ we set

$$\begin{aligned} \xi ^{\** }_{G^{\prime }, G}=\frac{c_1(G^{\prime })}{\mathop {\text {rk}}(\** )\mathop {\text {rk}}(G^{\prime })}-\frac{c_1(G)}{\mathop {\text {rk}}(\** )\mathop {\text {rk}}(G)}. \end{aligned}$$

We also set

$$\begin{aligned} K^+:=\{D\in \text {Num}(X)| D^2H^{d-2}>0, DH^{d-1}\geqslant 0 \text { for all nef }H\}. \end{aligned}$$

where $\text {Num}(X)={{\,\textrm{Pic}\,}}(X)\otimes {{\mathbb {R}}}/\sim$ and $\sim$ is an equivalence relation meaning $L_1\sim L_2$ if and only if $L_1AH^{d-2}=L_2AH^{d-2}$ for all divisors A on X.

Theorem A.4

If we have

$$\begin{aligned} H^{d}\cdot \Delta (E)\cdot H^{d-2}+\mathop {\text {rk}}(\** )^2\beta _{\mathop {\text {rk}}(E)}< 0, \end{aligned}$$

then there exists a saturated subsheaf $E^{\prime }\subset E$ such that $\xi ^{\** }_{E^{\prime }, E}\in K^{+}.$

We prove these theorems by induction on the rank $\mathop {\text {rk}}(E),$ and following Langer’s method. We only state the parts of the proof which are different to Langer’s method in smooth case and refer to [15, §3] for detailed arguments in the proof which is the same as Langer. For the induction process, let ${{\,\textrm{Thm}\,}}^{i}(\mathop {\text {rk}})$ represent the statement that Theorem A.i holds for ranks $\leqslant \mathop {\text {rk}}$ for $i=1,2,3,4.$ The rest of the induction chain follows the original proof due to Langer.

1.1 A.1. ${{\,\textrm{Thm}\,}}^{1}(\mathop {\text {rk}})$ implies ${{\,\textrm{Thm}\,}}^{2}(\mathop {\text {rk}})$

Suppose that the torsion free sheaf E is strongly modified slope semistable with respect to $(H, \** ),$ and $\Delta (E)\cdot H^{d-2}<0.$ We have $L_{\** ,\max }(E)=L_{\** ,\min }=\mu _{\** }(E).$ Theorem ${{\,\textrm{Thm}\,}}^{1}(\mathop {\text {rk}})$ implies that the restriction of E to H is still modified slope semistable. Since E is strongly modified slope semistable, $(F^k)^*E$ is also strongly modified slope semistable, and its restriction to a very general element in |H| is strongly modified slope semistable. Therefore by induction the restriction of $(F^k)^*E$ to a very general element in $H_1\cap \cdots \cap H_{d-1}$ for $H_1, \ldots , H_{d-1}\in |H|$ is strongly modified slope semistable. Therefore we are reduced to the two dimensional Deligne–Mumford stack case. Then this is Theorem 3.1.

1.2 A.2. ${{\,\textrm{Thm}\,}}^{2}(\mathop {\text {rk}})$ implies ${{\,\textrm{Thm}\,}}^{3}(\mathop {\text {rk}})$

First note that in this case, from (3.1), we have $\beta _{\mathop {\text {rk}}}.$ Our polarization is $(H, \** ),$ we first have the following inequality:

$$\begin{aligned}{} & {} H^{d}\cdot \Delta (E)H^{d-2}+\mathop {\text {rk}}(E)^2\mathop {\text {rk}}(\** )^2(L_{\** ,\max }(E)\nonumber \\{} & {} \quad -\mu _{\** }(E))(\mu _{\** }(E)-L_{\** ,\min }(E))\geqslant 0. \end{aligned}$$

(A.2)

To prove this inequality, first from the finite property fdHPN in Proposition 2.11 there exists a positive integer k such that all the quotients in the Harder–Narasimhan filtration of $(F^k)^*E$ are strongly modified slope semistable. Consider the Harder–Narasimhan filtration

$$\begin{aligned} 0=E_0\subset E_1\subset \cdots \subset E_m=(F^k)^*E \end{aligned}$$

and let $F_i=E_i/E_{i-1},$ $r_i=\mathop {\text {rk}}(F_i),$ $\mu _i=\mu _{\** }(F_i).$ The Hodge index theorem (holds for smooth Deligne–Mumford stacks and we give a short proof in Lemma A.5 below) implies that

$$\begin{aligned} \frac{\Delta ((F^k)^*E)H^{d-2}}{\mathop {\text {rk}}(E)}&=\sum _{i} \frac{\Delta (F_i)H^{d-2}}{r_i}-\frac{1}{\mathop {\text {rk}}(E)} \sum _{i<j}r_ir_j\left( \frac{c_1(F_i)}{r_i}-\frac{c_1(F_j)}{r_j}\right) ^2H^{d-2}\\&\geqslant \sum _{i}\frac{\Delta (F_i)H^{d-2}}{r_i}-\frac{\mathop {\text {rk}}(\** )^2}{H^{d}\mathop {\text {rk}}(E)} \sum _{i<j}r_ir_j(\mu _i-\mu _j)^2 \end{aligned}$$

${{\,\textrm{Thm}\,}}^{5}(\mathop {\text {rk}})$ implies that $\Delta (F_i)H^{d-2}\geqslant 0.$ Therefore by [15, Lemma 1.4], we have

$$\begin{aligned}{} & {} \frac{H^{d}\cdot \Delta (E)H^{d-2}}{\mathop {\text {rk}}(E)}\\{} & {} \quad \geqslant -\mathop {\text {rk}}(E)\mathop {\text {rk}}(\** )^2\left( \mu _{\** ,\max }((F^k)^*E)- \mu _{\** }((F^k)^*E)\right) \left( \mu _{\** }((F^k)^*E)-\mu _{\** ,\min }((F^k)^*E)\right) \end{aligned}$$

Both sides are divided by $p^{2k},$ we get:

$$\begin{aligned} H^{d}\cdot \Delta (E)H^{d-2}+\mathop {\text {rk}}(E)^2\mathop {\text {rk}}(\** )^2(L_{\** ,\max }(E)- \mu _{\** }(E))(\mu _{\** }(E)-L_{\** ,\min }(E))\geqslant 0. \end{aligned}$$

It is ready to prove ${{\,\textrm{Thm}\,}}^{3}(\mathop {\text {rk}}).$ Suppose that E is just modified slope semistable. We aim to use (A.2) and Corollary 2.10. The method is the same as in [15, §3.6], and we take an ample divisor D on X and set $H(t)=H+tD.$ Similar method shows that the Harder–Narasimhan filtration of E with respect to $(H(t), \** )$ is independent of t when t is positively small. Let $0=E_0\subset E_1\subset \cdots \subset E_m=E$ be the Harder–Narasimhan filtration with respect to $(H(t), \** ).$ We have (since E is modified slope semistable)

$$\begin{aligned} \mu _{\** , H}(E)\geqslant \mu _{\** , H}(E_1)=\lim _{t\rightarrow 0}\mu _{\** , H(t)}(E_1)\geqslant \lim _{t\rightarrow 0}\mu _{\** , H(t)}(E)=\mu _{\** ,H}(E). \end{aligned}$$

Hence

$$\begin{aligned} \lim _{t\rightarrow 0}\mu _{\** ,\max , H(t)}(E)=\lim _{t\rightarrow 0}\mu _{\** ,\max , H(t)}(E_1)=\mu _{\** ,H}(E), \end{aligned}$$

and similarly,

$$\begin{aligned} \lim _{t\rightarrow 0}\mu _{\** ,\min , H(t)}(E)=\mu _{\** ,H}(E). \end{aligned}$$

Thus we can apply (A.2) and Corollary 2.10, we get the result

$$\begin{aligned} H^{d}\cdot \Delta (E)H^{d-2}+\mathop {\text {rk}}(\** )^2\beta _{\mathop {\text {rk}}(E)}\geqslant 0. \end{aligned}$$

Lemma A.5

Let $\pi : \mathcal {X}\rightarrow X$ be the coarse moduli associated to a tame smooth projective surface Deligne–Mumford stack $\mathcal {X}$ over an algebraically closed field k. Consider $\pi ^{*}H$ where H is an ample divisor on X and let ${{\mathcal {D}}}$ be a numerically non trivial divisor with ${{\mathcal {D}}}\cdot \pi ^{*}H=0,$ then ${{\mathcal {D}}}^{2}<0.$

Proof

First we can assume that $\mathcal {X}$ is a quotient stack because of projectivity. From [6, Corollary 4.4], there exists a finite surjective schematic cover of degree d given by $p:Y\rightarrow \mathcal {X}.$ Since $\pi$ is finite (as the stack has finite diagonal) we have $f=\pi \circ p$ is a finite morphism to X and hence Y is projective. From the proof of the same corollary we also have Y can be taken to be smooth. Hence Y is a smooth projective surface and Hodge Index Theorem is known for Y. Next we have $f^{*}H$ is ample and $p^{*}[{{\mathcal {D}}}]$ is the class of a divisor induced in $A^{*}(Y).$ Also note $p^{*}[{{\mathcal {D}}}]\cdot f^{*}H=d [{{\mathcal {D}}}]\cdot \pi ^{*}H=0$ from our assumption. Hence $(p^{*}[{{\mathcal {D}}}])^{2}=d[{{\mathcal {D}}}]^{2}$ which is less than 0 from Hodge Index for smooth projective surfaces and hence we have the desired result. $\square$

1.3 A.3. ${{\,\textrm{Thm}\,}}^{3}(\mathop {\text {rk}})$ implies ${{\,\textrm{Thm}\,}}^{4}(\mathop {\text {rk}})$

If we have the condition in ${{\,\textrm{Thm}\,}}^{4}(\mathop {\text {rk}}),$ i.e., $H^{d}\cdot \Delta (E)H^{d-2}+\mathop {\text {rk}}(\** )^2\beta _{\mathop {\text {rk}}(E)}< 0.$ Then from ${{\,\textrm{Thm}\,}}^{3}(\mathop {\text {rk}}),$ E is not modified slope semistable. Let $E^{\prime }\subset E$ be the maximal destabilizing subsheaf of E and set $E^{\prime \prime }=E/E^{\prime },$ $r^{\prime }=\mathop {\text {rk}}(E^{\prime }),$ $r^{\prime \prime }=\mathop {\text {rk}}(E^{\prime \prime }).$ First we calculate:

$$\begin{aligned} \frac{\Delta (E)H^{d-2}}{\mathop {\text {rk}}(E)} +\frac{r r^{\prime }\xi ^2_{E^{\prime }, E}H^{d-2}}{r^{\prime \prime }}=\frac{\Delta (E^{\prime })H^{d-2}}{r^{\prime }}+ \frac{\Delta (E^{\prime \prime })H^{d-2}}{r^{\prime \prime }}. \end{aligned}$$

We also have $\mathop {\text {rk}}(\** )^2\frac{\beta _{\mathop {\text {rk}}(E)}}{\mathop {\text {rk}}(E)}\geqslant \mathop {\text {rk}}(\** )^2(\frac{\beta _{r^{\prime }}}{r^{\prime }}+\frac{\beta _{r^{\prime \prime }}}{r^{\prime \prime }}).$ Since $H^{d}\cdot \Delta (E)H^{d-2}+\mathop {\text {rk}}(\** )^2\beta _{\mathop {\text {rk}}(E)}< 0,$ and we require $H^d>0,$ either $\xi ^2_{E^{\prime }, E}>0$ or at least one of the $H^{d}\cdot \Delta (E^{\prime })H^{d-2}+\mathop {\text {rk}}(\** )^2\beta _{r^{\prime }}$ and $H^{d}\cdot \Delta (E^{\prime \prime })H^{d-2}+\mathop {\text {rk}}(\** )^2\beta _{r^{\prime \prime }}$ is negative. Therefore the same argument as in [8, Theorem 7.3.3] gives the result.

1.4 A.4. ${{\,\textrm{Thm}\,}}^{4}(\mathop {\text {rk}})$ implies ${{\,\textrm{Thm}\,}}^{2}(\mathop {\text {rk}})$

Suppose that $\Delta (E)H^{d-2}<0.$ The condition in ${{\,\textrm{Thm}\,}}^{4}(\mathop {\text {rk}}),$ applying to $(F^l)^*E$ (since E is strongly modified slope semistable by the condition in ${{\,\textrm{Thm}\,}}^{2}(\mathop {\text {rk}})$), is:

$$\begin{aligned} H^{d}\cdot \Delta ((F^l)^*E)H^{d-2}+\mathop {\text {rk}}(\** )^2\beta _{\mathop {\text {rk}}(E)}< 0 \end{aligned}$$

which is equivalent to (logarithm with base p)

$$\begin{aligned} l>\frac{1}{2}\log _{p}\left( -\frac{\mathop {\text {rk}}(\** )^2\beta _{\mathop {\text {rk}}(E)}}{H^{d}\cdot \Delta (E)H^{d-2}}\right) . \end{aligned}$$

Then for large l,  there exists a saturated torsion free subsheaf $E^{\prime }\subset (F^l)^*E$ such that $\xi _{E^{\prime }, (F^l)^*E}\in K^+.$ By “self-duality” property of $K^+$ we have $\xi _{E^{\prime }, (F^l)^*E}H^{d-1}>0,$ which means that the sheaf E is not strongly modified semistable, a contradiction.

1.5 A.5. ${{\,\textrm{Thm}\,}}^{2}(\mathop {\text {rk}}-1)$ implies ${{\,\textrm{Thm}\,}}^{1}(\mathop {\text {rk}})$

We use $\Pi =|H|$ to denote the complete linear system, and let $Z:=\{(D,x)\in \Pi \times X: x\in D\}$ be the incidence variety. Let

$$\begin{aligned} p: Z\rightarrow \Pi ; \quad q: Z\rightarrow X \end{aligned}$$

be the corresponding projections. For each $s\in \Pi ,$ let $Z_s$ be the scheme theoretic fiber of p over the point s. Consider the following cartesian diagram:

Then ${{\mathcal {Z}}}$ is a Deligne–Mumford stack which is given by $\{(\pi ^{-1}(D),x): (D,x)\in \Pi \times X, x\in D\}.$ The pull-back of the generating sheaf $\**$ on $\mathcal {X}$ under q gives the generating sheaf $q^*\**$ on ${{\mathcal {Z}}}.$

We work on the sheaf $q^*E$ for a torsion free sheaf E on $\mathcal {X},$ and let

$$\begin{aligned} 0=E_0\subset E_1\subset \cdots \subset E_m=q^*E \end{aligned}$$

be the modified relative Harder–Narasimhan filtration with respect to $p\circ \pi$ and the pull-back generating sheaf. This means that there exists an open subset $U\subset \Pi$ such that all $F_i=E_i/E_{i-1}$ are flat over U and for each $s\in U$ the fibers $(E_{\bullet })_s$ is the modified Harder–Narasimhan filtration of $E_s=q^*E|_{{{\mathcal {Z}}}_s}$ for ${{\mathcal {Z}}}_s=\pi ^{-1}(Z_s).$ As in the proof of [15, §3.9], the modified relative Harder–Narasimhan filtration is actually the Harder–Narasimhan filtration of $q^*E$ with respect to

$$\begin{aligned} ((\pi ^*p^*{{\mathcal {O}}}_{\Pi }(1))^{\dim (\Pi )}q^*H, q^*\** ). \end{aligned}$$

By the finite property in Sect. 2.4, for the sheaf $q^*E,$ there exists a positive integer k such that all the quotients in the Harder–Narasimhan filtration of $(F^k)^*(q^*E)=q^*((F^k)^*E)$ are strongly modified semistable. We will prove the inequality (A.1), and from [15, Lemma 1.5], when applying to the polygons of the Harder–Narasimhan filtration for modified slopes, we just prove the case that all the graded pieces $F_i$’s are strongly modified slope semistable with respect to $(p^*{{\mathcal {O}}}_{\Pi }(1)^{\dim (\Pi )}q^*H, q^*\** ).$

We perform the same argument as in [15, §3.9], and let $\Lambda \subset \Pi$ be a pencil. Set $Y=p^{-1}(\Lambda ),$ and ${{\mathcal {Y}}}=(p\circ \pi )^{-1}(\Lambda )\subset {{\mathcal {Z}}}.$ From the same arguments as in [15, §3.9], we have $q|_{{{\mathcal {Y}}}}: {{\mathcal {Y}}}\rightarrow \mathcal {X}$ to be the stacky blow up of $\mathcal {X}$ along ${{\mathcal {B}}}$ which is the base locus of the pull back of the general pencil $\pi ^{-1}(\Lambda ).$ If $d\geqslant 3,$ then ${{\mathcal {B}}}$ is a smooth connected substack (this follows from the following Lemma A.6 that we prove below), and there is only one exceptional divisor ${{\mathcal {N}}}$ for $q|_{{{\mathcal {Y}}}}.$ We write down

$$\begin{aligned} c_1(F_i|_{Y})=q|_{{{\mathcal {Y}}}}^*{{\mathcal {M}}}_i+ b_i {{\mathcal {N}}}\end{aligned}$$

where ${{\mathcal {M}}}_i$ are divisors on $\mathcal {X}$ which are pullbacks of divisors $M_i$ on X and $b_i$ are rational numbers. If the dimension $d=2,$ then B consists of $N=H^d$ distinct points and ${{\mathcal {B}}}$ consists of N distinct stacky points. Let ${{\mathcal {N}}}_1,\cdots , {{\mathcal {N}}}_{N}$ be the exceptional divisors of $q|_{{{\mathcal {Y}}}}.$ There exist rational numbers $b_{ij}$ and divisors ${{\mathcal {M}}}_i$ such that

$$\begin{aligned} c_1(F_i|_{Y})=q|_{{{\mathcal {Y}}}}^*{{\mathcal {M}}}_i+ \sum _{j}b_{ij} {{\mathcal {N}}}_j. \end{aligned}$$

Let $b_i=(\sum _{j}b_{ij})/N.$ We have

$$\begin{aligned} \mu _{\** ,i}=\frac{c_1(F_i|_{Y})p^*{{\mathcal {O}}}_{\Pi }(1)q^*H^{d-2}}{r_i\mathop {\text {rk}}(\** )}=\frac{{{\mathcal {M}}}_i\cdot H^{d-1}+b_iN}{r_i\mathop {\text {rk}}(\** )}. \end{aligned}$$

${{\,\textrm{Thm}\,}}^{2}(\mathop {\text {rk}}-1)$ implies that $\Delta (F_j|_{Y})p^*{{\mathcal {O}}}_{\Pi }(1)q^*H^{d-2}\geqslant 0$ for every j. We calculate

$$\begin{aligned}&\frac{N\Delta (E)q|_{{{\mathcal {Y}}}}^*H^{d-2}}{\mathop {\text {rk}}(E)}=\sum _{i} \frac{N\Delta (F_i|_{{{\mathcal {Y}}}})q|_{{{\mathcal {Y}}}}^*H^{d-2}}{r_i}\\&\qquad -\frac{N}{\mathop {\text {rk}}(E)}\sum _{i<j}r_ir_j\left( \frac{c_1(F_i|_{{{\mathcal {Y}}}})}{r_i}- \frac{c_1(F_j|_{{{\mathcal {Y}}}})}{r_j}\right) ^2\cdot q|_{{{\mathcal {Y}}}}^*H^{d-2}\\&\quad \geqslant \frac{N}{\mathop {\text {rk}}(E)}\sum _{i<j}r_ir_j\left( N\left( \frac{b_i}{r_i}- \frac{b_j}{r_j}\right) ^2-\left( \frac{{{\mathcal {M}}}_i}{r_i}-\frac{{{\mathcal {M}}}_j}{r_j}\right) ^2H^{d-2}\right) \\&\quad \geqslant \frac{1}{\mathop {\text {rk}}(E)}\sum _{i<j}r_ir_j\left( (N)^2\left( \frac{b_i}{r_i}- \frac{b_j}{r_j}\right) ^2-\left( \frac{{{\mathcal {M}}}_iH^{d-1}}{r_i}- \frac{{{\mathcal {M}}}_jH^{d-1}}{r_j}\right) ^2\right) . \end{aligned}$$

The last inequality is from Hodge index theorem Lemma A.5 for smooth Deligne–Mumford stacks, and from the slope $\mu _{\** ,i},$ the last expression above gives

$$\begin{aligned} 2\sum _{i}N b_i\mu _{\** , i}-\frac{1}{\mathop {\text {rk}}(E)}\sum _{i<j}r_i r_j (\mu _{\** ,i}-\mu _{\** ,j})^2. \end{aligned}$$

To prove the claim, first $(q|_{{{\mathcal {Y}}}})_*(E_i|_{{{\mathcal {Y}}}})\subset E$ implies that

$$\begin{aligned} \frac{\sum _{j\leqslant i}{{\mathcal {M}}}_jH^{d-1}}{\mathop {\text {rk}}(\** )\sum _{j\leqslant i}r_j}\leqslant \mu _{\** , \max }(E) \end{aligned}$$

which gives the inequality:

$$\begin{aligned} \sum _{j\leqslant i}b_j N\geqslant \sum _{j\leqslant i}\mathop {\text {rk}}(\** )r_j(\mu _{\** , j}-\mu _{\** , \max }(E)). \end{aligned}$$

(A.3)

Therefore

$$\begin{aligned}&\sum _{i}Nb_i \mu _{\** ,i}=\sum _{i}\left( \sum _{i<j}Nb_j\right) (\mu _{\** ,i}-\mu _{\** ,i+1})\\&\quad \geqslant \sum _{i}\left( \sum _{j\leqslant i}\mathop {\text {rk}}(\** )r_j(\mu _{\** , j}- \mu _{\** , \max }(E))\right) (\mu _{\** ,i}-\mu _{\** ,i+1})\\&\quad =\mathop {\text {rk}}(\** )\cdot \sum _{i<j}\frac{r_ir_j}{\mathop {\text {rk}}(E)} (\mu _{\** ,i}-\mu _{\** ,j})^2+\mathop {\text {rk}}(E)(\mu _{\** }(E)- \mu _{\** ,\max }(E))(\mu _{\** }(E)-\mu _{\** ,\min }(E)). \end{aligned}$$

So we get:

$$\begin{aligned}&\frac{N\Delta (E)q|_{{{\mathcal {Y}}}}^*H^{d-2}}{\mathop {\text {rk}}(E)}\\&\quad \geqslant \sum _{i<j}\frac{2\mathop {\text {rk}}(\** )-1}{\mathop {\text {rk}}(E)}r_i r_j(\mu _{\** ,i}-\mu _{\** ,j})^2+2\mathop {\text {rk}}(E)(\mu _{\** }(E)- \mu _{\** ,\max }(E))(\mu _{\** }(E)-\mu _{\** ,\min }(E)). \end{aligned}$$

Lemma A.6

Let $\pi : \mathcal {X}\rightarrow X$ be the coarse moduli associated to a tame smooth projective Deligne–Mumford stack $\mathcal {X}$ over an algebraically closed field k of dimension greater than 2. Let |D| be the complete linear system associated to a very ample divisor on X,  and let B be the base locus of a general pencil associated to |D|,  then ${{\mathcal {B}}}:=B\times _{X} \mathcal {X}$ is smooth and connected substack of $\mathcal {X}.$

Proof

Let $p:Y\rightarrow \mathcal {X}$ be a projective smooth scheme with p being faithfully flat and finite from [14, Theorem 2.1] and $f:Y\rightarrow X$ be the composition $\pi \circ p,$ which is finite. We note that $f^{*}D$ is a base point free divisor on Y since f is finite. Then if $\Lambda$ be a sufficiently general pencil associated to |D| with base locus B then $f^{-1}(\Lambda )$ is a general pencil associated with the base point free linear system $|f^{*}D|$ with base locus given by $f^{-1}(B)$ which from Bertini’s Theorem on smooth projective varieties is smooth and connected. Thus $f:f^{-1}(B)\rightarrow B$ being surjective we obtain B is connected and ${{\mathcal {B}}}$ being the stack with coarse moduli B we have the connectedness of ${{\mathcal {B}}}.$

To prove smoothness of ${{\mathcal {B}}}$ we note that $p:f^{-1}(B)\rightarrow {{\mathcal {B}}}$ is a faithfully flat and finite morphism and hence we conclude the smoothness of ${{\mathcal {B}}}$ from the descent of smoothness of $f^{-1}(B)$ along p, see [27]. $\square$

1.6 A.6. Restriction theorems

We generalize the restriction theorem of Langer to smooth Deligne–Mumford stacks in higher dimensions. We give a general statement for the (A.2). Recall from Corollary 2.10,

$$\begin{aligned} \alpha _{\** }(E):=\max (L_{\** ,\max }(E)-\mu _{\** , \max }(E), \mu _{\** , \min }(E)-L_{\** ,\min }(E)). \end{aligned}$$

Let A be a nef divisor for X such that $\pi _*(T_{\mathcal {X}}\otimes \** )(A)$ is globally generated and a $\pi$ ample line bundle ${{\mathcal {L}}},$ then we have,

$$\begin{aligned} \alpha _{\** }(E)\leqslant \frac{\mathop {\text {rk}}(E)-1}{p-1}\left( max_{1\leqslant k\leqslant m}\mu _{\** }(A\otimes {{\mathcal {L}}}^{k})\right) . \end{aligned}$$

See Corollary 2.10 for the details of this inequality.

Theorem A.7

Let E be a torsion free sheaf on a smooth Deligne–Mumford stack $\mathcal {X}.$ Then we have

$$\begin{aligned}{} & {} H^{d}\cdot \Delta (E)H^{d-2}+\mathop {\text {rk}}(E)^2\mathop {\text {rk}}(\** )^2(L_{\** ,\max }(E)\nonumber \\{} & {} \quad -\mu _{\** }(E))(\mu _{\** }(E)-L_{\** ,\min }(E))\geqslant 0 \end{aligned}$$

(A.4)

and

$$\begin{aligned}{} & {} H^{d}\cdot \Delta (E)H^{d-2}+\mathop {\text {rk}}(E)^2\mathop {\text {rk}}(\** )^2(\mu _{\** ,\max }(E)\nonumber \\{} & {} \quad -\mu _{\** }(E))(\mu _{\** }(E)-\mu _{\** ,\min }(E))+\beta _{r}\geqslant 0. \end{aligned}$$

(A.5)

Proof

The proof of (A.4) is the same as in Claim (A.2). The proof of formula (A.5) is the same as [15, Theorem 5.1]. $\square$

Theorem A.8

Let E be a torsion free sheaf of rank $\mathop {\text {rk}}(E)\geqslant 2$ on a smooth Deligne–Mumford stack $\mathcal {X}.$ Suppose that E is slope modified stable with respect to $(\** , {{\mathcal {O}}}_{X}(1)=H).$ Let $D\subset |mH|$ be a normal divisor such that $E|_{{{\mathcal {D}}}}$ has no torsion where ${{\mathcal {D}}}=\pi ^{-1}(D)).$ If

$$\begin{aligned} m>\lfloor \frac{\mathop {\text {rk}}(E)-1}{\mathop {\text {rk}}(E)\mathop {\text {rk}}(\** )}\Delta (E)H^{d-2}+\frac{1}{H^d \mathop {\text {rk}}(E)\mathop {\text {rk}}(\** )(\mathop {\text {rk}}(E)-1)}+\frac{(\mathop {\text {rk}}(E)-1)\beta _{r}}{H^d \mathop {\text {rk}}(E)\mathop {\text {rk}}(\** )} \rfloor \end{aligned}$$

then $E|_{{{\mathcal {D}}}}$ is slope modified stable with respect to $(\** |_{{{\mathcal {D}}}}, H|_{{{\mathcal {D}}}}).$

Proof

The proof is the same as [15, Theorem 5.2]. $\square$

Appendix B: Bogomolov’s inequality for Higgs sheaves

By Yunfeng Jiang, Promit Kundu and Hao (Max) Sun

Let k be an algebraically closed field of characteristic $p\geqslant 0$ and ${\mathcal {X}}$ be a smooth tame projective Deligne–Mumford stack of dimension d over k with coarse moduli space $\pi :{\mathcal {X}}\rightarrow X.$ Let $\**$ be a generating sheaf on ${\mathcal {X}}$ satisfying Condition $\star$ in Sect. 3.1, and let H an ample divisor on X.

Definition B.1

A Higgs sheaf $(E, \theta )$ is a pair consisting of a coherent sheaf $E\in \hbox {Coh}({\mathcal {X}})$ and an ${\mathcal {O}}_{{\mathcal {X}}}$-homomorphism $\theta : E\rightarrow E\otimes \Omega _{{\mathcal {X}}}$ satisfying the integrability condition $\theta \wedge \theta =0.$ We say a Higgs sheaf $(E, \theta )$ a system of Hodge sheaves if there is a decomposition $E=\oplus E^i$ such that $\theta : E^i\rightarrow E^{i-1}\otimes _{{\mathcal {O}}_{{\mathcal {X}}}}\Omega _{{\mathcal {X}}}$

• (1) We say that $(E, \theta )$ is slope semistable if $\mu _{\** }(E^{\prime })\leqslant \mu _{\** }(E)$ for every Higgs subsheaf $(E^{\prime },\theta ^{\prime })$ of $(E,\theta ),$ which means that $E^{\prime }\subset E$ is a subsheaf and $\theta ^{\prime }: E^{\prime }\rightarrow E^{\prime }\otimes \Omega _{\mathcal {X}}$ is preserved by $\theta .$

• (2) A system of Hodge sheaves $(E, \theta )$ is slope semistable if the inequality $\mu _{\** }(E^{\prime })\leqslant \mu _{\** }(E)$ is satisfied for every subsystem of Hodge sheaves $(E^{\prime },\theta ^{\prime })$ of $(E,\theta ).$ Here a subsystem means that $E^{\prime }\subset E$ is a subsheaf, and $E^{\prime }=\oplus E^{\prime i}$ such that $\theta ^{\prime }: E^{\prime i}\rightarrow E^{\prime i}\otimes _{\mathcal {X}}\Omega _{\mathcal {X}}$ is preserved by $\theta .$

We recall the main results of Ogus and Vologodsky [24], where the theory is for schemes, but in étale topology it works for Deligne–Mumford stacks.

Assume that $p>0.$ Let S be a scheme over k and $f:{\mathcal {X}}\rightarrow S$ be a morphism of stacks over k. A lifting of ${\mathcal {X}}/S$ modulo $p^2$ is a morphism ${\widetilde{f}}: \widetilde{{\mathcal {X}}}\rightarrow {\widetilde{S}}$ of flat ${\mathbb {Z}}/p^2{\mathbb {Z}}$-stacks such that f is the base change of ${\widetilde{f}}$ by the closed embedding $S\rightarrow {\widetilde{S}}$ defined by p. Let $\hbox {MIC}_{p-1}({\mathcal {X}}/S)$ be the category of ${\mathcal {O}}_{{\mathcal {X}}}$-modules with an integrable connection whose p-curvature is nilpotent of level $\leqslant p-1.$ Here we say a ${\mathcal {O}}_{\mathcal {X}}$-modules V with integrable connection such that the p-curvature $\Psi : T_{\mathcal {X}/S}\rightarrow {\mathcal {E}}nd_S(V)$ is nilpotent of level $\leqslant p-1$ if for all open subsets $U\subset \mathcal {X}$ and for all derivations $D\in T_{\mathcal {X}/S}(U)$ we have $\Psi (D)^p=0.$

Let $\hbox {HIG}_{p-1}({\mathcal {X}}^{(1)}/S)$ denote the category of Higgs ${\mathcal {O}}_{{\mathcal {X}}^{(1)}}$-modules with a nilpotent Higgs field of level $\leqslant p-1,$ which means that $\theta ^p=0.$ We have the following theorem of Ogus and Vologodsky [24, Theorem 2.8].

Theorem B.2

If $f:{\mathcal {X}}\rightarrow S$ is a smooth morphism with a lifting $\widetilde{{\mathcal {X}}}^{(1)}\rightarrow {\widetilde{S}}$ of ${\mathcal {X}}^{(1)}\rightarrow S$ modulo $p^2,$ then the Cartier transform

$$\begin{aligned} C_{{\mathcal {X}}/S}:\hbox {MIC}_{p-1}({\mathcal {X}}/S) \rightarrow \hbox {HIG}_{p-1}({\mathcal {X}}^{(1)}/S) \end{aligned}$$

defines an equivalence of categories with quasi-inverse

$$\begin{aligned} C^{-1}_{{\mathcal {X}}/S}:\hbox {HIG}_{p-1}({\mathcal {X}}^{(1)}/S) \rightarrow \hbox {MIC}_{p-1}({\mathcal {X}}/S). \end{aligned}$$

Lemma B.3

Let $(E,\theta )\in \hbox {HIG}_{p-1}({\mathcal {X}}^{(1)}/S).$ Then we have $[C^{-1}_{{\mathcal {X}}/S}(E)]=F_g^*[E],$ where $[\cdot ]$ denotes the class of a coherent sheaf in the Grothendieck group $K_0({\mathcal {X}}).$

Proof

See [16, Lemma 2]. $\square$

Corollary B.4

Assume $S=\mathop {\text {Spec}}\nolimits k,$ and let $(E,\theta )\in \hbox {HIG}_{p-1}({\mathcal {X}}^{(1)}/S).$ Then $(E, \theta )$ is slope semistable with respect to $(H,\** )$ iff $C^{-1}_{{\mathcal {X}}/S}(E)$ is slope $\nabla$-semistable with respect to $(F_g^*H,F_g^*\** ).$

Proof

The proof is the same as that of [16, Corollary 1]. $\square$

Lemma B.5

Let $(E, \theta )$ be a torsion free slope semistable Higgs sheaf on ${\mathcal {X}}.$ Then there exists an ${\mathbb {A}}^1$-flat family of Higgs sheaves $({\widetilde{E}}, {\widetilde{\theta }})$ on ${\mathcal {X}}\times {\mathbb {A}}^1$ such that the restriction $({\widetilde{E}}_t,{\widetilde{\theta }}_t)$ to the fiber over any closed point $t\in {\mathbb {A}}^1$ is isomorphic to $(E,\theta )$ and $(E_0, \theta _0)$ is a slope semistable system of Hodge sheaves.

Proof

See [17, Corollary 5.7]. $\square$

Proposition B.6

Let V be a torsion free sheaf on ${\mathcal {X}},$ then

$$\begin{aligned} H^{d-2}\Delta (V) \geqslant -\frac{(\mathop {\text {rank}}\nolimits V\mathop {\text {rank}}\nolimits \** )^2}{H^d}(L_{\** ,\max }(V)-\mu _{\** }(V))(\mu _{\** }(V)-L_{\** ,\min }(V)). \end{aligned}$$

Proof

The proposition follows from Theorem A.2 by the same arguments as in the proof of Theorem A.3. $\square$

Theorem B.7

Assume $p=0.$ Let $(E, \theta )$ be a slope semistable Higgs sheaf with respect to $(H,\** ).$ Then we have $H^{d-2}\Delta (E)\geqslant 0.$

Proof

Deforming $(E,\theta )$ to a system of Hodge sheaves (see Lemma B.5) we can assume that $(E, \theta )$ is nilpotent. This is because that if $(E=\oplus _{j=1}^n E^j, \theta )$ is the Hodge sheaf. From $\theta : E^j\rightarrow E^{j-1}\otimes \Omega _{\mathcal {X}},$ one sees that $\theta ^n=0,$ i.e., $(E, \theta )$ is nilpotent. Now we use the standard reduction to positive characteristic technique, which we recall for the convenience of the reader (see [21, section 2]). There exists a finitely generated ${\mathbb {Z}}$-algebra $R\subset k$ and a tame smooth Deligne–Mumford stack $\widetilde{{\mathcal {X}}}\rightarrow S=\mathop {\text {Spec}}\nolimits R$ such that ${\mathcal {X}}=\widetilde{{\mathcal {X}}}\times _S\mathop {\text {Spec}}\nolimits k.$ Let ${\widetilde{\pi }}: \widetilde{{\mathcal {X}}}\rightarrow {\widetilde{X}}$ be its coarse moduli space. We can assume that $X={\widetilde{X}}\times _S\mathop {\text {Spec}}\nolimits k,$ $\pi$ is induced by ${\widetilde{\pi }}$ after base change, and there exists an ample divisor ${\widetilde{H}}$ on ${\widetilde{X}}$ extending H and a generating sheaf ${\widetilde{\** }}$ on $\widetilde{{\mathcal {X}}}$ extending $\**$ such that it is a direct sum of $\pi$ ample vector bundles obtained from direct sum of line bundles. Since we have the Higgs sheaf on the generic fiber, which can be extended to a general fiber. Thus, we can also assume there exists an S-flat family of Higgs sheaves $({\widetilde{E}}, {\widetilde{\theta }})$ on $\widetilde{{\mathcal {X}}}$ extending $(E, \theta ).$

Shrinking S,  by openness of semistability we can assume that $({\widetilde{E}}_s, {\widetilde{\theta }}_s)$ is slope semistable with respect to $({\widetilde{\** }}_s, {\widetilde{H}}_s)$ for any $s\in S.$ Choose a closed point $s\in S$ such that the characteristic q of the residue field k(s) is $\geqslant \mathop {\text {rank}}\nolimits E.$ We can choose such a closed point from the following reason. For a big enough prime number q,  one takes the prime ideal generated by q in R. Then one sees that the characteristic of the residue field of that ideal is q. Then the stack $\widetilde{{\mathcal {X}}}\times _S\mathop {\text {Spec}}\nolimits (R/m_s^2)$ is a lifting of $\widetilde{{\mathcal {X}}}_s$ modulo $q^2.$ By Corollary B.4, one can associate to $({\widetilde{E}}_s, {\widetilde{\theta }}_s)$ a slope $\nabla$-semistable sheaf with integrable connection $(V_s, \nabla _s)$ with respect to $(F_g^*{\widetilde{H}}_s,F_g^*{\widetilde{\** }}_s).$ By Lemma B.3, one sees that $[V_s]=[C^{-1}_{X/S}(E_s)]=[F^*_g(E_s)].$ Hence we have $ch_0(V_s)=ch_0(E_s),$ $ch_1(V_s)=q ch_1(E_s)$ and $ch_2(V_s)=q^2 ch_2(E_s).$ These imply the following equality,

$$\begin{aligned} {\widetilde{H}}_s^{d-2}\Delta (V_s)=q^2{\widetilde{H}}_s^{d-2}\Delta ({\widetilde{E}}_s). \end{aligned}$$

(B.1)

Let $0=V_0\subset V_1\subset \cdots \subset V_m=V_s$ be the usual Harder–Narasimhan filtration of $V_s,$ then by [9, Lemma 2.7], the induced morphisms $V_i\rightarrow (V_s/V_i)\otimes \Omega _{\widetilde{{\mathcal {X}}}_s}$ are nonzero ${{\mathcal {O}}}_{\widetilde{{\mathcal {X}}}_s}$-morphisms. Take a nef divisor A on ${\widetilde{X}}_s$ such that $\pi _*(T_{\widetilde{{\mathcal {X}}}_s}\otimes {\widetilde{\** }}_s)(A)$ is globally generated. From [9, Proposition 2.10 and Corollary 2.11], it follows that

$$\begin{aligned} \mu _{{\widetilde{\** }}_s, \max }(V_s)-\mu _{{\widetilde{\** }}_s, \min }(V_s)\leqslant (\mathop {\text {rank}}\nolimits V_s-1)\left( \frac{{\widetilde{H}}_s^{d-1}A}{{\widetilde{H}}_s^d}+M\right) \end{aligned}$$

and

$$\begin{aligned} \max (L_{{\widetilde{\** }}_s, \max }(V_s)-\mu _{{\widetilde{\** }}_s, \max }(V_s), \mu _{{\widetilde{\** }}_s,\min }(V_s)-L_{{\widetilde{\** }}_s, \min }(V_s)) \leqslant \frac{\mathop {\text {rank}}\nolimits V_s-1}{q-1}\left( \frac{{\widetilde{H}}_s^{d-1}A}{{\widetilde{H}}_s^d}+M\right) , \end{aligned}$$

for some constant M depending on A and $\** .$ They imply that

$$\begin{aligned} (L_{{\widetilde{\** }}_s,\max }(V_s)-L_{{\widetilde{\** }}_s,\min }(V_s))\leqslant \frac{\mathop {\text {rank}}\nolimits V_s-1}{1-\frac{1}{q}}\left( \frac{{\widetilde{H}}_s^{d-1}A}{{\widetilde{H}}_s^d}+M\right) . \end{aligned}$$

Hence Proposition B.6 gives

$$\begin{aligned} q^2{\widetilde{H}}_s^{d-2}\Delta ({\widetilde{E}}_s)={\widetilde{H}}_s^{d-2}\Delta (V_s) \geqslant -\frac{(\mathop {\text {rank}}\nolimits V_s\mathop {\text {rank}}\nolimits \** )^2}{{\widetilde{H}}_s^d}(\frac{\mathop {\text {rank}}\nolimits V_s-1}{1-\frac{1}{q}})^2\left( \frac{{\widetilde{H}}_s^{d-1}A}{{\widetilde{H}}_s^d}+M\right) ^2. \end{aligned}$$

Taking sufficiently large q,  one obtains

$$\begin{aligned} H^{d-2}\Delta (E)={\widetilde{H}}_s^{d-2}\Delta ({\widetilde{E}}_s)\geqslant 0. \end{aligned}$$

$\square$

Using the same argument in [16, section 6], one can recover [4, Theorem 1.1]:

Theorem B.8

Assume that $p=0,$ $d=2,$ and the canonical line bundle $K_{{\mathcal {X}}}$ is nef,  then we have

$$\begin{aligned} c^2_1(T_{{\mathcal {X}}})\leqslant 3c_2(T_{{\mathcal {X}}}). \end{aligned}$$