Boundedness of Semistable Sheaves

Introduction

The existence of moduli space is one of the oldest and hottest topic in algebraic geometry. We are interested in if there exist a space that “parametrize” desired geometric objects, and if it exists, whether it’s proper or not. The moduli space $\mathfrak{M}$ that parametrize coherent sheaves over algebraic varieties $X$ encodes a lot of geometric information about $X$ . However, such $\mathfrak{M}$ does not always exist. Sometimes the family we want to parametrize is too “big” to make $\mathfrak{M}$ into a scheme, so we may focus on a family $\mathfrak{F}$ with specific property. In our case, we want to know some information about moduli space of torsion-free semistable sheaves with fixed Hilbert polynomial. To construct such a moduli space, the first and the most fundamental problem is the boundedness of the family $\mathfrak{F}$ , i.e. the moduli space, if it exists, whether it is of finite type over the base field.

We will establish the bounded of torsion-free semistable sheaves with algebraically closed base field with characteristic zero following steps in Hu&Lehn⁷. We will begin with an introduction of $\mu$ -semistable sheaves and a complete proof of Kleiman’s criterion, which is an important criterion for boundedness. The second result is Grauert-Mülich Theorem. It describes the behaviour of semistable sheaf $\mathcal{F}$ under restriction on general $\mathcal{F}$ -regular sequences. Then we will discuss semistability of tensor product of sheaves and use it to prove the Le Potier-Simpson Theorem. We will also mention some approaches to the same problem in positive characteristic case in the last section.

Preliminaries

In this section, we set up the preliminaries for the semistable sheaves and boundedness of sheaves. For the part of of semistabilities, we refer Hu&Lehn⁷ for details and proofs. We will provide a useful criterion Theorem 17.

Let $X$ be a projective variety over some algebraically closed field $k$ . Fix an ample line sheaf $\mathcal{O}_{X}(1)$ and the corresponding ample divisor class $H$ . For any coherent sheaf $\mathcal{F}$ , the dimension of $\mathcal{F}$ is the dimension of $\mathop{\mathrm{Supp}}\mathcal{F}$ , which is exactly the degree of Hilbert polynomial $P(\mathcal{F})(m)$ with the respect to $H$ . $P(\mathcal{F})(m)$ can be expressed into the form $\sum\limits_{i=0}^{\dim\mathcal{F}}\alpha_{i}(\mathcal{F})\frac{m^{i}}{i!}$ with rational coefficients.

Definition 1. Let $\mathcal{F}$ be a coherent sheaf of dimension $d=\dim(X)$ , the degree of $\mathcal{F}$ is defined by

\deg(\mathcal{F}):= \alpha_{d-1}(\mathcal{F})-\mathop{\mathrm{rk}}(\mathcal{F})\cdot \alpha_{d-1}(\mathcal{O}_{X})

and its slope by

\mu(\mathcal{F}):=\frac{\deg(\mathcal{F})}{\mathop{\mathrm{rk}}(\mathcal{F})},

and the slope of Hilbert polynomial by

\hat{\mu}(\mathcal{F}):=\frac{\alpha_{d-1}(\mathcal{F})}{\alpha_{d}(\mathcal{F})}.

Clearly, the relation of $\mu(\mathcal{F})$ and $\hat{\mu(\mathcal{F})}$ can be given by $\mu(\mathcal{F})=\alpha_{d}(\mathcal{O}_{X})\hat{\mu(\mathcal{F})}-\alpha_{d-1}(\mathcal{O}_{X})$ . For smooth $X$ , Hirzebruch-Riemann-Roch Theorem shows that $\deg \mathcal{F}=c_{1}(\mathcal{F})\cdot H^{d-1}$ , which is an integer.

Definition 2. Let $\mathcal{F}$ be a coherent sheaf of dimension $d=\dim(X)$ . $\mathcal{F}$ is called $\mu$ -semistable (or just semistable) if any subsheaf $\mathcal{E}$ has dimension less than $d$ , then $\mathcal{E}$ has dimension less than $d-1$ , and for all subsheaf $\mathcal{G}\subset \mathcal{F}$ with $0<\mathop{\mathrm{rk}}\mathcal{G}<\mathop{\mathrm{rk}}\mathcal{F}$ , we have $\mu(\mathcal{G})\leq \mu(\mathcal{F})$ . $\mathcal{F}$ is called $\mu$ -stable if the above inequality is strict.

For torsion-free sheaf $\mathcal{F}$ , the definition is equivalent to say: $\mathcal{F}$ is (semi)stable if for all subsheaf $\mathcal{G}$ of $\mathcal{F}$ with $0<\mathop{\mathrm{rk}}(\mathcal{G})<\mathop{\mathrm{rk}}(\mathcal{F})$ , $\mu(\mathcal{G})< \mu(\mathcal{F})$ (resp. $\mu(\mathcal{G})\leq \mu(\mathcal{F})$ ). We will mainly focus on torsion-free semistable sheaves. An equivalent definition can be state in terms of quotient sheaves:

There’s another stablity called “Gieseker stablity”. The stablity involves pure sheaf:

Definition 3. A coherent sheaf $\mathcal{F}$ is pure of dimension $d$ if $\dim \mathcal{F}=d$ and all nontrivial coherent subsheaf $\mathcal{E}$ of $\mathcal{F}$ has dimension $d$ .

Clearly, a sheaf $\mathcal{F}$ is pure of dimension $\dim X$ is equivalent to $\mathcal{F}$ is torsion free. Gieseker stablity encodes more information than $\mu$ -stablity. The coherent sheaf $\mathcal{F}$ is (semi)stable if $\mathcal{F}$ is pure and for all proper subsheaf $\mathcal{E}$ , on has $\frac{P(\mathcal{E})(m)}{\alpha_{\dim \mathcal{E}}(\mathcal{E})} (\leq )\frac{P(\mathcal{F})(m)}{\alpha_{\dim \mathcal{F}}(\mathcal{F})}$ .

Lemma 4. Let $\mathcal{F}$ be a coherent sheaf. Then $\mathcal{F}$ is semistable if and only if for all quotient sheaves $\mathcal{E}$ of $\mathcal{F}$ , $\mu(\mathcal{E})\geq \mu(\mathcal{F})$ .

Proof

Proof. See Hu&Lehn⁷ section 1.2. ◻

The following lemma is useful in constructing Harder-Narasimhan filtration:

Lemma 5. Let $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ be semistable sheaves with $\mu(\mathcal{F}_{1})>\mu(\mathcal{F}_{2})$ , then $\mathop{\mathrm{Hom}}(\mathcal{F}_{1},\mathcal{F}_{2})=0$ .

Proof

Proof. Loc.cit. ◻

The followings are some basic facts about semistability

Remark 6.

A line sheaf $\mathcal{L}$ is stable.
Let $\mathcal{F}$ be a semistable sheaf and $\mathcal{L}$ a line sheaf, then $\mathcal{F}\otimes \mathcal{L}$ is semistable.
Let $0\to \mathcal{F}'\to \mathcal{F}\to \mathcal{F}''\to 0$ be a short exact sequence of sheaves. Then
If $\mathcal{F}$ is semistable and either $\mu(\mathcal{F})=\mu(\mathcal{F}')$ or $\mu(\mathcal{F})=\mu(\mathcal{F}'')$ , then $\mathcal{F}'$ and $\mathcal{F}''$ are semistable.
If $\mathcal{F}'$ and $\mathcal{F}''$ are semistable and $\mu(\mathcal{F}')=\mu(\mathcal{F}'')$ , then $\mathcal{F}$ is
semistable.

A torsion-free sheaf $\mathcal{F}$ may not be semistable, but $\mathcal{F}$ always admits a Harder-Narasimhan filtration whose factors are semistable. Harder-Narasimhan filtration is a very important tool in study of semistable sheaves.

Definition 7. Let $\mathcal{F}$ be a coherent sheaf on $X$ . A Harder-Narasimhan filtration is an increasing filtration

0=HN_{0}(\mathcal{F})\subset HN_{1}(\mathcal{F})\subset\cdots\subset HN_{l}(\mathcal{F})=\mathcal{F},

such that the factors $gr_{i}^{HN}=HN_{i}(\mathcal{F}) /HN_{i-1}(\mathcal{F})$ for $i=1,\dots, l$ , are semistable sheaves with slope $\mu_{i}$ , satisfying

\mu_{max}:=\mu_{1}>\mu_{2}>\cdots>\mu_{l}=:\mu_{min}.

We call $gr_{1}^{HN}$ the maximal destablizing sheaf and $gr_{l}^{HN}$ the minimal destablizing quotient. One can generalize Lemma 5 to torsion-free sheaves using notation above:

Lemma 8. Let $\mathcal{F}$ and $\mathcal{G}$ be torsion-free sheaves and $\mu_{min}(\mathcal{F})>\mu_{max}(\mathcal{G})$ , then $\mathop{\mathrm{Hom}}(\mathcal{F},\mathcal{G})=0$ .

Theorem 9. Every torsion-free sheaf $\mathcal{F}$ has a unique Harder-Narasimhan filtration. Moreover, all factors of the Harder-Narasimhan filtration is torsion-free.

Proof

Proof. The idea is to find a subsheaf $\mathcal{E}$ of $\mathcal{F}$ such that for all subsheaf $\mathcal{G}\subset \mathcal{F}$ , one has $\mu(\mathcal{E})\geq \mu(\mathcal{G})$ , and moreover, $\mu(\mathcal{E})=\mu(\mathcal{G})$ only if $\mathcal{G}\subset\mathcal{E}$ . Then we have $\mathcal{E}$ unique and semistable. For more details, see Hu&Lehn⁷ section 1.3. ◻

Harder-Narasimhan filtration can be done on a family of sheaves parametrized on some integral scheme. For the proof we again refer to Hu&Lehn⁷ section 2.3. We explain some notation that will be used in the articles here:

For $f:X\to S$ a morphism of finite type noetherian schemes an $g:T\to S$ , the notation $X_{T}$ will be used for the fibre product $T\times_{S} X$ , and $g_{X}:X_{T}\to X$ , $f_{T}:X_{T}\to T$ are the natural projections. For $s\in S$ and coherent sheaf $\mathcal{F}$ on $X$ , $X_{s}$ denotes the fibre $f^{-1}(s)=\mathop{\mathrm{Spec}}k(s)\times_{S}X$ and $\mathcal{F}_{s}=\mathcal{F}|_{X_{s}}$ .

Theorem 10. *Let $S$ be a finite type integral scheme over $k$ . Let $f:X\to S$ bea projective morphism and $\mathcal{O}_{X}(1)$ be an $f$ -ample line sheaf on $X$ . Let $\mathcal{F}$ be a flat family of coherent sheaves on the fibre of closed point on $S$ . There is a projective birational morphism $g:T\to S$ of integral $k$ -scheme and a filtration

0=HN_{0}(\mathcal{F})\subset HN_{1}(\mathcal{F})\subset\cdots\subset HN_{l}(\mathcal{F})=g_{X}^{*}\mathcal{F}

such that the following holds:*

The factors $HN_{i}(\mathcal{F}) /HN_{i-1}(\mathcal{F})$ are $T$ -flat
for all $i=1,\dots, l$ ;
There is a dense open subscheme $U\subset T$ such that
$HN_{i}(\mathcal{F})_{t}=g_{X}^{*}HN_{i}(\mathcal{F}_{g(t)})$ for all $t\in U$ . Here $HN_{i}(\mathcal{F}_{(g(t))})$ is the Harder-Narasimhan filtration of $\mathcal{F}_{g(t)}$ .

Remark 11.

$\mu_{min}(\mathcal{F}\oplus \mathcal{E})=\min\{\mu_{min}(\mathcal{F}),\mu_{min}(\mathcal{E})\}$ ;
$\mu_{max}(\mathcal{F}\oplus \mathcal{E})=\max\{\mu_{max}(\mathcal{F}),\mu_{max}(\mathcal{E})\}$ .
$\mu(\mathcal{F}\otimes \mathcal{E})=\mu(\mathcal{F})+\mu(\mathcal{E})$ .
$\mu_{min}(\mathcal{F}\otimes \mathcal{E})\leq \mu_{min}(\mathcal{F})+\mu_{min}(\mathcal{E})$ , $\mu_{max}(\mathcal{F}\otimes \mathcal{E})\geq \mu_{max}(\mathcal{F})+\mu_{max}(\mathcal{E})$ .
For exact sequence $0\to \mathcal{F}'\to \mathcal{F}\to \mathcal{F}''$ , $\mu_{min}(F)\geq \mu_{min}(\mathcal{F}'')$ and $\mu_{max}(\mathcal{F})\geq \mu_{max}(\mathcal{F}')$ .

To construct a moduli space of sheaves, we need to make sure that our family of sheaves is not too big to parametrize. We will prove this is true for semistable sheaves on $k$ variety with $\mathop{\mathrm{char}}k=0$ in this report. We introduce the idea of boundedness and several criterion of boundedness next. The notation of Castelnuovo-Mumford regularity is needed.

Definition 12. Let $m\in \mathbb{Z}$ and $\mathcal{F}$ coherent sheaf on $X$ . $\mathcal{F}$ is $m$ -regular if $H^{i}(X,\mathcal{F}(m-i))=0$ for all $i>0$ .

Lemma 13. If $\mathcal{F}$ is $m$ -regular, then the following facts holds:

$\mathcal{F}$ is $m'$ -regular for all $m'\geq m$ .
$\mathcal{F}(m)$ is generated by global sections.
The natural evaluation homomorphism $H^{0}(X,\mathcal{F}(m))\otimes H^{0}(X,\mathcal{O}(n))\to H^{0}(X,\mathcal{F}(m+n))$ are surjective for all $n\geq 0$ .

Proof

Proof. Follow the idea in Mum¹². Without losing of generality we can assume $X=\mathbb{P}^{d}$ . $k$ is algebraically closed so we can find a hyperplane section $H$ which does not contain any of the associated points of $\mathcal{F}$ . Then the sequence $$0\to \mathcal{F}(n-1)\to\mathcal{F}(n)\to \mathcal{F}_{H}(n)\to 0$$ is exact.

For 1, the long exact sequence gives $H^{i}(X,\mathcal{F}(n-i-1))\to H^{i}(X,\mathcal{F}(n-i))\to H^{i}(H,\mathcal{F}_{H}(n-i))$ . Proceeding a induction on $n$ , we may assume $H^{i}(X,\mathcal{F}(n-i-1))=0$ . Similarly, by induction on the dimension $d$ , we may assume $H^{i}(H,\mathcal{F}_{H}(n))=0$ . So $H^{i}(X,\mathcal{F}(n-i))=0$ . This shows $\mathcal{F}$ is $m'$ -regular for $m'\geq m$ .

For 3, we only prove the case $n=1$ , the general cases are similar. We use induction on $d$ . Consider the commutative diagram

The top morphism is the tensor product of two surjective morphism so it’s surjective. By inductive hypothesis, $\tau$ is also surjective, so $\nu\circ \mu$ is surjective and $H^{0}(X,\mathcal{F}(m+1))=\mathop{\mathrm{im}}\mu+\mathop{\mathrm{ker}}\nu$ . The bottom row is exact, we also have $H^{0}(X,\mathcal{F}(m+1))=\mathop{\mathrm{im}}\mu+\mathop{\mathrm{im}}\alpha$ . Note that $\alpha$ is given by tensoring the local section defining $H$ , so $\mathop{\mathrm{im}}\alpha\subset \mathop{\mathrm{im}}\mu$ . So $H^{0}(X,\mathcal{F}(m+1))=\mathop{\mathrm{im}}\mu$ .

For 2, take sufficiently large $n$ , $\mathcal{F}(n)$ is generated by global sections. Note $H^{0}(X,\mathcal{F}(m))\otimes H^{0}(X,\mathcal{O}(n-m))\to H^{0}(X,\mathcal{F}(n))$ is surjective, $\mathcal{F}(n)$ is generated by $H^{0}(X,\mathcal{F}(m))\otimes H^{0}(X,\mathcal{O}(n-m))$ . At any point $p\in X$ , the local case of above surjection shows that $\mathcal{F}(m)_{p}=\mathcal{F}(n)_{p}$ is generated by $H^{0}(X,\mathcal{F}(m))$ . So $\mathcal{F}(m)$ is generated by $H^{0}(X,\mathcal{F}(m))$ . ◻

Thanks to 1 in Lemma 13, we can define the regularity of a coherent sheaf $\mathcal{F}$ to be $\mathop{\mathrm{reg}}(\mathcal{F}):=\inf\{m\in \mathbb{Z}|\mathcal{F}\text{ is } m\text{-regular}\}$ . Now we give the definition for boundedness of a family of sheaves.

Definition 14. A family $\mathfrak{F}$ of isomorphism class of coherent sheaves on $X$ is bounded if there is a $k$ -scheme $S$ of finite type and a coherent sheaf $\mathcal{F}$ on $X\times S$ , such that $\mathfrak{F}$ is the subset of $\{\mathcal{F}_{s} | s\text{ is closed point in }S\}$ .

Lemma 15. The following property of a family of sheaves $\{\mathcal{F}_{i}\}_{i\in I}$ are equivalent:

The family is bounded.
The set of Hilbert polynomial $\{P(\mathcal{F}_{i})(m)\}_{i\in I}$ is finite and there is a uniform bound $\mathop{\mathrm{reg}}(\mathcal{F}_{i})\leq \rho$ for all $i\in I$ .
The set of Hilbert polynomial $\{P(\mathcal{F}_{i})(m)\}_{i\in I}$ is finite and there is a coherent sheaf $\mathcal{F}$ such that all $\mathcal{F}_{i}$ admit a surjection $\mathcal{F}\to \mathcal{F}_{i}$ for all $i\in I$ .

Proof

Proof. 1 $\Rightarrow$ 2: The finiteness of Hilbert polynomial is from the flatten stratification lemma (c.f. Hu&Lehn⁷ section 2.1). For the regularity part, note $S$ is quasicompact and we may reduce to the case $S$ is affine. There is a $m>0$ such that $H^{i}(X\times S,\mathcal{F}(n))=0$ for all $i>0$ and $n>m$ . On the fibre $H^{i}(X,\mathcal{F}|_{\mathop{\mathrm{Spec}}k(s)\times X}(m+d-i))=H^{i}(X\times S,\mathcal{F})\times k(s)=0$ , here $d=\dim X$ . Thus $\mathop{\mathrm{reg}}(\mathcal{F}_{i})\leq m+d$ .

2 $\Rightarrow$ 3: Lemma 13 shows $\mathcal{F}_{i}(\rho)$ are generated by global sections, so there’s surjections $\mathcal{O}(-\rho)^{m}\to \mathcal{F}_{i}$ with $m\geq \max\{P(\mathcal{F}_{i})(\rho)\}$ . We will need the finiteness of the set $\{P(\mathcal{F}_{i})(m)\}$ here.

3 $\Rightarrow$ 1: There’re only finitely many Hilbert polynomials. Let $S=\coprod\mathrm{Quot}_{P(\mathcal{F}_{i})}$ be the disjoint union of the Quot scheme corresponding to those finitely many Hilbert polynomials. Then 1 is immediate from the definition of Quot scheme. ◻

The following proposition allows us to estimate the regularity of a coherent sheaf $\mathcal{F}$ in terms of Hilbert polynomial and the number of global sections of the restriction of $\mathcal{F}$ to regular sequence of hyperplane sections.

Proposition 16. There’re universal polynomials $P_{i}\in \mathbb{Q}[T_{0},\dots, T_{i}]$ such that the following holds: Let $\mathcal{F}$ be a coherent sheaf of dimension $\dim(\mathcal{F})\leq d$ and let $H_{1},\dots, H_{d}$ be an regular sequence of hyperplane sections. If $\chi(\mathcal{F}|_{\cap_{j\leq i}H_{j}})=a_{i}$ and $h^{0}(\mathcal{F}|_{\cap_{j\leq i}}H_{j})\leq b_{i}$ , then $$\mathop{\mathrm{reg}}(\mathcal{F})\leq P_{d}(a_{0}-b_{0},\dots, a_{d}-b_{d}).$$

Proof

Proof. It suffices to show for the case $X=\mathbb{P}^{d}$ and $\dim\mathcal{F}=d$ . By the argument of Lemma 1.2.1 in Hu&Lehn⁷, the Hilbert polynomial $P(\mathcal{F})(m)$ can be written into $\sum\limits_{i=0}^{d}a_{i}\binom{m+i-1}{i}$ . The proof proceed by induction on the dimension of the sheaf.

The base case is clear: for zero dimension sheaf, $P_{0}$ can be taken as any polynomial.

Let $d\geq 1$ , take any hyperplane scetion $H$ which does not meet any associated points of $\mathcal{F}$ , we have the exact sequence

0\to \mathcal{F}(m-1)\to \mathcal{F}(m)\to \mathcal{F}_{H}(m)\to 0

and the long exact sequence

\cdots\to H^{i}(X,\mathcal{F}(m-1))\to H^{i}(X,\mathcal{F}(m))\to H^{i}(H, \mathcal{F}_{H}(m))\to H^{i+1}(X,\mathcal{F}(m-1))\to \cdots

By induction hypothesis, $\mathcal{F}|_{H}$ is $n=P_{d-1}(a_{1}-b_{1},\dots, a_{d}-b_{d})$ -regular. For $m\geq n-1$ , the long exact sequence and Lemma 13 shows that $H^{i}(X,\mathcal{F}(m))\cong H^{i}(X,\mathcal{F}(m-1))$ for all $i\geq 2$ , $m\geq n-2$ . For sufficiently large $m$ , the cohomologies vanishes so all $H^{i}(X,\mathcal{F}(m))=0$ for $i\geq 2$ , $m\geq n-2$ . We also get a surjection

\nu: H^{1}(X,\mathcal{F}(m-1))\to H^{1}(X,\mathcal{F}(m)),

the function $h^{1}(\mathcal{F}(m))$ is decreasing in $m$ . $\nu$ becomes an isomorphism if and only if the homomorphism $H^{0}(X,\mathcal{F}(m))\to H^{0}(H,\mathcal{F}_{H}(m))$ is surjective. Use the same diagram as in the proof of Lemma 13, we can conclude that if $H^{0}(X,\mathcal{F}(m))\to H^{0}(H,\mathcal{F}_{H}(m))$ is surjective, then $H^{0}(X,\mathcal{F}(m+1))\to H^{0}(H,\mathcal{F}_{H}(m+1))$ is surjective. Once $h^{1}(\mathcal{F}(m))=h^{1}(\mathcal{F}(m+1))$ , the value never decreases anymore. So $h^{1}(\mathcal{F}(m))$ strictly decrease to 0. For $m\geq n+h^{1}(\mathcal{F}(n))+1$ , $H^{1}(X,\mathcal{F}(m-1))=0$ .

Now we estimate the upper bound for $h^{1}(\mathcal{F}(n))$ by a polynomial in $a_{i}-b_{i}$ .

h^{1}(\mathcal{F}(n))=h^{0}(\mathcal{F}(n))-\chi(\mathcal{F}(n))=h^{0}(\mathcal{F}(n))-\sum\limits_{i=0}^{d}a_{i}\binom{n+i-1}{i}

and

h^{0}(\mathcal{F}(n))\leq \sum_{i=0}^{d}b_{i}\binom{n+i-1}{i}

can be done inductively:

h^{0}(\mathcal{F}(n))\leq h^{0}(\mathcal{F}(n-1))+h^{0}(\mathcal{F}_{H}(n))

and $h^{0}(\mathcal{F}_{H}(n))\leq \sum\limits_{i=1}^{d}b_{i}\binom{n+i-2}{i-1}$ implies $h^{0}(\mathcal{F}(n))\leq \sum\limits_{i=0}^{d}b_{i}\binom{n+i-1}{i}$ . So $h^{1}(\mathcal{F}(n))\leq \sum\limits_{i=0}^{d}(a_{i}-b_{i})\binom{n+i-1}{i}$ , where $n=P_{d-1}(a_{1}-b_{1},\dots, a_{d}-b_{d})$ . We may take $P_{d}(a_{0}-b_{0},\dots, a_{d}-b_{d}) = n+\sum\limits_{i=0}^{d}(a_{i}-b_{i})\binom{n+i-1}{i}$ . ◻

Combining Lemma 13 and Proposition 16, we get the important criterion for boundedness.

Theorem 17 (Kleiman Criterion). Let $\{\mathcal{F}_{i}\}$ be a family of coherent sheaf on $X$ with the same Hilbert polynomial $P$ . Then this family is bounded if and only if there are constants $C_{i}$ , $i=0,\dots, d=\deg(P)$ , such that for every $\mathcal{F}_{i}$ there exists an $\mathcal{F}_{i}$ regular sequence of hyperplane sections $H_{1},\dots, H_{d}$ such that $h^{0}(\mathcal{F}|_{\cap j\leq i})\leq C_{i}$ .

Example 18. Let $X$ be a smooth projective curve over algebraically closed field $k$ . If $\{\mathcal{F}_{i}\}$ is a family of coherent sheaves with $h^{0}(\mathcal{F}_{i})$ and $\mathop{\mathrm{rk}}(\mathcal{F}_{i})$ , then $\{\mathcal{F}_{i}\}$ is bounded. This is because $h^{0}(\mathcal{F}_{i}|_{H})=\mathop{\mathrm{rk}}(\mathcal{F})\cdot \deg X$ is bounded.

If $\{\mathcal{F}_{i}\}$ is a family of semistable sheaves with fixed Hilbert polynomial, then $\{\mathcal{F}_{i}\}$ is bounded.

Grauert-Mülich Theorem

To make a use of Kleiman criterion, we first need to understand the behaviour of semistable sheaves under the restriction to some hypersurface sections. Although for general hypersurface $H$ , the restriction of semistable sheaf $\mathcal{F}$ may not be semistable, $\mathcal{F}|_{H}$ cannot be so ‘far’ from semistable. The first thing we need to prove is Grauert-Mülich theorem. We begin with some set up on incidence structures.

Let $k$ be algebraically field of characteristic zero. Let $X$ be a normal projective variety over $k$ of $\dim X\geq 2$ and fix a very ample sheaf $\mathcal{O}(1)$ on $X$ . Denote the linear system $\left| \mathcal{O}(a) \right|$ by $\Pi_{a}$ and let $Z_{a}=\{(D,x)\in \Pi_{a}\times X|x\in D\}$ be the incidence variety. We also allow the incidence structures on different linear systems: Let $\Pi=\Pi_{a_{1}}\times \cdots\times \Pi_{a_{l}}$ and $Z=Z_{a_{1}}\times_{X}\cdots\times_{X} Z_{a_{l}}$ . Then there are natural projection $p:Z\to \Pi$ and $q:Z\to X$ .

Let $V_{a}=H^{0}(X,\mathcal{O}(a))$ and $\mathcal{K}$ be the kernel of the natural evaluation map $V_{a}\otimes \mathcal{O}_{X}\to \mathcal{O}_{X}(a)$ . Then $Z_{a}=\mathbb{P}(\check{\mathcal{K}})$ and there’s a natural closed immersion $Z_{a}\to \mathbb{P}(\check{V})\times X$ . $q$ is the bundle morphism and therefore open. We can compute the relation tangent bundle by Euler sequence:

0\to \mathcal{O}_{Z_{a}}\to q^{*}\mathcal{K}\otimes p^{*}\mathcal{O}(1)\to \mathcal{T}_{Z_{a}/X}\to 0

$Z$ parametrizes the intersection of element in $\Pi_{a_{i}}$ in such a way: For $s=(s_{1},s_{2},\dots, s_{l})$ be a closed point in $\Pi$ . Each $s_{i}$ corresponds to a divisor $D_{i}$ . Then the fibre $p^{-1}(s)$ can be identified by $q$ with the scheme-theoretic intersection $D_{1}\cap D_{2}\cap \cdots\cap D_{l}\subset X$ . The relative tangent bundle of $Z$ can be similarly computed as $\mathcal{T}_{Z/X}=p_{1}^{*}\mathcal{T}_{Z_{a_{1}} /X}\oplus\cdots\oplus p_{l}^{*}\mathcal{T}_{Z_{a_{l}} /X}$ , where $p_{i}:Z\to Z_{a_{i}}$ are the natural projections. For a coherent sheaf $\mathcal{F}$ on $X$ , let $\mathcal{E}=q^{*}\mathcal{F}$ , from the construction we have $\mathcal{E}_{s}=\mathcal{F}|_{Z_{s}}$ .

Lemma 19. Let $\mathcal{F}$ be a torsion free coherent sheaf on $X$ and $\mathcal{E}\cong q^{*}\mathcal{F}$ . Then

There is a nonempty open subset $S'\subset \Pi$ such that the morphism $p_{S'}:Z_{S'}\to S'$ is flat and for all $s\in S'$ , the fibre $Z_{s}=p^{-1}(s)$ is a normal irreducible complete intersection of codimension $l$ in $X$ .

There is a nonempty dense open subset $S\subset S'$ such that the family $\mathcal{E}_{S}=q^{*}\mathcal{F}|_{Z_{S}}$ is flat over $S$ and for all $s\in S$ , the fibre $\mathcal{E}_{s}$ is torsion free.

Proof

Proof. The flatness is a generic condition. The remaining part follows from Bertini theorem, see Ha⁶ section 2.8.

The flatness is same as above. We reduce to the case of $X$ smooth first. Let $f:X\to \mathbb{P}^{n}$ be the closed immersion defining $\mathcal{O}(1)$ . We can regard $\mathcal{F}$ as a pure sheaf of dimension $\dim X$ supported on $X$ . Let $S''$ be the open subset which contains point $s$ that parametrizes regular sequence for $\mathcal{F}$ and $\mathcal{E}xt^{i}(\mathcal{F},\omega_{X})$ , $\forall i\geq 0$ . Then clearly $S''$ is not empty. Let $S=S''\cap S$ . On each $Z_{s}=p^{-1}(s)$ , the torsion-freeness is from the following lemma:

Lemma 20 (Hu&Lehn⁷). Let $X$ be a smooth projective variety over $k$ . For a coherent sheaf $\mathcal{F}$ of codimension $c$ , we say $\mathcal{F}$ satisfies Serre condition $S_{k,c}$ if $\mathop{\mathrm{depth}}\mathcal{F}_{x}\geq \min\{k,\dim\mathcal{O}_{X,x}-c\}$ for all $x\in \mathop{\mathrm{Supp}}(\mathcal{F})$ . Then

$\mathcal{F}$ is pure if and only if $\mathcal{F}$ satisfies $S_{1,c}$ .
Let $H$ be a hypersurface defined by some ample line sheaf $\mathcal{L}$ . If $H$ is a $\mathcal{F}$ regular section and $\mathcal{F}$ satisfies $S_{k,c}$ , then $\mathcal{F}|_{H}$ satisfies $S_{k-1,c+1}$ .

Use the lemma and induction, one can easily see $\mathcal{F}|_{Z_{s}}$ satisfies $S_{1,n-\dim X+l}$ , which means $\mathcal{F}|_{Z_{s}}$ is pure of dimension $\dim X-l$ and thus torsion free on $Z_{s}$ . ◻

We may shrink to a smaller open dense set $S$ such that all the factors are flat. $S$ is irreducible and thus connected, so $\mu((\mathcal{E}_{i}/\mathcal{E}_{i-1})_{s})$ is a constant for each $s\in S$ . We may define $\mu_{i}=\mu((\mathcal{E}_{i}/\mathcal{E}_{i-1})_{s})$ . Then

\mu_{1}>\mu_{2}>\cdots>\mu_{j-1}.

Define the number of gap by

\delta\mu=\left\{\begin{aligned} &0 &&\text{ if }\mathcal{E}_{s} \text{ is semistable}\\ &\max\{\mu_{i}-\mu_{i+1}\} &&\text{ otherwise} \end{aligned}\right.

The Grauert-Mülich theorem gives us an upper bound $\delta\mu$ for sufficient general $s\in S$ .

Theorem 21 (Grauert-Mülich). Let $\mathcal{F}$ be a semistable torsion-free sheaf. Then there is a nonempty open dense subset $S$ of $\Pi$ such that for all $s\in S$ , the following inequality holdes:

\delta\mu(\mathcal{F}|_{Z_{s}})\leq \max\{a_{i}\}\cdot\prod_{i=1}^{l} a_{i}\cdot \deg X.

Proof

Proof. The case $\mathcal{E}_{s}$ is semistable is trivial. Assume $\delta\mu>0$ and $\delta \mu= \mu_{i}-\mu_{i+1}$ for specific $i$ and let $\mathcal{E}'=\mathcal{E}_{i}$ , $\mathcal{E}''=\mathcal{E} /\mathcal{E}'$ . For all $s\in S$ , $\mathcal{E}_{s}'$ and $\mathcal{E}_{s}''$ are torsion free and from the uniqueness of Harder-Narasimhan filtration, $\mu_{min}(\mathcal{E}_{s}')=\mu_{i}$ and $\mu_{max}(\mathcal{E}_{s}'')=\mu_{i+1}$ .

Since torsion-free sheaves are locally free on a open subset, we may let $Z_{0}$ be the maximal open subset of $Z_{S}$ such that $\mathcal{E}|_{Z_{0}}$ and $\mathcal{E}''|_{Z_{0}}$ are locally free. Let their ranks be $r$ and $r''$ , respectively. The surjection $\mathcal{E}_{Z_{0}}\to \mathcal{E}_{Z_{0}}''$ gives a morphism $\varphi:Z_{0}\to \mathop{\mathrm{Grass}}(\mathcal{F},r'')$ , and $\mathcal{E}|_{Z_{0}}\to \mathcal{E}|_{Z_{0}}$ is the pullback of $\mathcal{F}\to \mathcal{U}$ . Let $X_{0}$ be the image of $Z_{0}$ in $X$ , since $q:Z\to X$ is the bundle morphism, $X_{0}$ is open. Note $\mathcal{F}$ is torsion free, for any $s\in S$ the complement of $Z_{0}\cap Z_{s}$ in $Z_{s}$ has codimension larger than $1$ , the codimension of complement of $X_{0}$ in $X$ is also larger than $1$ .

Let

D\varphi:\mathcal{T}_{Z/X}|_{Z_{0}}\to \varphi^{*}\mathcal{T}_{\mathop{\mathrm{Grass}}(\mathcal{F},r'') /X}

be the relative differential morphism related to $\varphi$ . Since $\mathcal{T}_{\mathop{\mathrm{Grass}}(\mathcal{F},r'') /X}=\mathop{\mathcal{H}om}(\mathop{\mathrm{ker}}(\mathcal{F}\to \mathcal{U}),\mathcal{U})$ , we can identify $\varphi^{*}\mathcal{T}_{\mathop{\mathrm{Grass}}(\mathcal{F},r'') /X}$ as $\mathop{\mathcal{H}om}(\varphi^{*}\mathop{\mathrm{ker}}(\mathcal{F}\to \mathcal{U}),\varphi^{*}\mathcal{U})=\mathop{\mathcal{H}om}(\mathcal{E}|_{Z_{0}}',\mathcal{E}|_{Z_{0}}'')$ . Thus $D\varphi$ corresponds to

\Phi:(\mathcal{E}'\otimes \mathcal{T}_{Z /X})|_{Z_{0}}\to \mathcal{E}|_{Z_{0}}''

via the isomorphism $\mathop{\mathrm{Hom}}((\mathcal{E}'\otimes \mathcal{T}_{Z /X})|_{Z_{0}},\mathcal{E}|_{Z_{0}}'')\cong \mathop{\mathrm{Hom}}(\mathcal{T}_{Z /X}|_{Z_{0}},\mathop{\mathcal{H}om}(\mathcal{E}|_{Z_{0}}',\mathcal{E}|_{Z_{0}}''))$ .

Next we want to show $\Phi_{s}\neq 0$ for general $s\in S$ . Suppose on the contrary. We may shrink $S$ smaller if necessary to make $\Phi=0$ . Since $q$ is faithfully flat, according to Theorem 40, $\mathcal{E}|_{X_{0}}$ is also locally free. Restricting $\varphi$ to $Z_{0}$ , we have the following diagram:

$q_{0}$ is a smooth morphism with connected fibres. If $\Phi=0$ , then $D\varphi=0$ and in characteristic zero case, this will imply $\varphi$ is constant on fibre of $q_{0}$ . Then by rigidity lemma, there is a morphism $\rho:X_{0}\to \mathop{\mathrm{Grass}}(\mathcal{F}|_{X_{0}},r'')$ . From the universal property of $\mathop{\mathrm{Grass}}(\mathcal{F}|_{X_{0}},r'')$ , there’s a quotient $\mathcal{F}|_{X_{0}}\to \mathcal{F}''$ of rank $r''$ . Moreover, $\mathcal{F}|_{X_{0}\cap Z_{s}}''\cong \mathcal{E}''_{s}|_{Z_{0}\cap Z_{s}}$ for general $s\in S$ . Since $\mathcal{F}|_{X_{0}}$ is support on codimension $\geq 2$ sets in $X$ , any extension $\mathcal{F}'''$ of $\mathcal{F}''$ to $X$ satisfies $\mu(\mathcal{F}''')=\mu(\mathcal{F}'')$ . By our assumption, $\mathcal{E}_{s}''$ is a destablizing quotient of $\mathcal{E}_{s}$ , this means $\mathcal{F}'''$ is destablizing quotient of $\mathcal{F}$ . This contradicts the assumption that $\mathcal{F}$ is semistable.

A nontrivial $\Phi_{s}$ defines a morphism $\mathcal{E}_{s}'\otimes \mathcal{T}_{Z /X}|_{Z_{s}}$ to $\mathcal{E}_{s}''$ in the quotient category $\mathop{\mathrm{Coh}}_{n-l,n-l-1}(Z_{s})$ . Then by Lemma 5, we have the inequality

\mu_{min}(\mathcal{E}_{s}'\otimes \mathcal{T}_{Z /X}|_{Z_{s}})\leq \mu_{max}(\mathcal{E}_{s}'').

Consider the Koszul complex associated to the evaluation map $e:V_{a}\otimes \mathcal{O}_{X}\to \mathcal{O}(a)$ . Taking the last terms, we get a surjection $\wedge^{2} V_{a}\otimes \mathcal{O}_{X}(-a)\to \mathop{\mathrm{ker}}(e)=\mathcal{K}$ and hence a surjection

\wedge^{2}V_{a}\otimes q^{*}\mathcal{O}_{X}(-a)\otimes p^{*}\mathcal{O}(1)\to q^{*}\mathcal{K}\otimes p^{*}\mathcal{O}(1)\to \mathcal{T}_{Z_{a} /X}.

Using $\mathcal{T}_{Z /X}=\bigoplus\limits_{i} p_{i}^{*}\mathcal{T}_{Z_{a_{i}} /X}$ , we have a surjection

(\bigoplus_{i}\wedge^{2} V_{a_{i}}\otimes \mathcal{O}(-a_{i}))|_{Z_{s}}\to \mathcal{T}_{Z /X}|_{Z_{s}}.

Therefore,

\begin{aligned} \mu_{min}(\mathcal{E}_{s}'\otimes\mathcal{T}_{Z /X}|_{Z_{s}})&\geq \mu_{min}(\bigoplus_{i}\wedge^{2}V_{a_{i}}\otimes \mathcal{O}(-a_{i})\otimes \mathcal{E}'|_{Z_{s}})\\&=\min\limits_{i}\{\mu_{min}(\mathcal{O}_{Z_{s}(-a_{i})}\otimes \mathcal{E}_{s}')\}\\&=\mu_{min}(\mathcal{E}_{s}')-\max\{a_{i}\}\cdot\deg(Z_{s}).\end{aligned}

In conclusion,

\begin{aligned} \delta\mu=\mu_{i}-\mu_{i+1}&=\mu_{min}(\mathcal{F}_{s}')-\mu_{max}(\mathcal{F}_{s}'')\\&\leq \max\{a_{i}\}\cdot \deg(Z_{s})\\&=\max\{a_{i}\}\cdot \prod a_{i}\cdot \deg(X).\end{aligned}

◻

Corollary 22. Let $\mathcal{F}$ be a torsion-free semistable sheaf of rank $r$ on $X$ . Let $Y$ be the intersection of $s<\dim X$ general hyperplanes in $\left| \mathcal{O}_{X}(1) \right|$ . Then

\mu_{min}(\mathcal{F}|_{Y})\geq \mu(\mathcal{F})-\frac{r-1}{2}\deg(X)

and

\mu_{max}(\mathcal{F}|_{Y})\leq \mu(\mathcal{F})+\frac{r-1}{2}\deg(X).

Proof

Proof. We only show the inequality for $\mu_{max}$ , $\mu_{min}$ is similar. If $\mathcal{F}|_{Y}$ is semistable then there’s nothing to prove. Let $\mu_{1},\dots, \mu_{j}$ and $r_{1},\dots,r_{j}$ be the slopes and ranks of the factors of the Harder-Narasimhan filtration of $\mathcal{F}|_{Y}$ . Note all the $r_{i}$ are positive integers since the factors are torsion-free. By Theorem 21, we have $$\mu_{i}-\mu_{i+1}\leq \deg X.$$ Take the sum from 1 to $i$ we get $\mu_{i}\geq \mu_{1}-(i-1)\deg X$ . So

\begin{aligned} \mu(\mathcal{F})&= \sum_{i=1}^{j}\frac{r_{i}}{r}\mu_{i}\\& \geq \mu_{1}-\frac{\deg X}{r}\sum_{i=1}^{j}(i-1)r_{i}\\& \geq \mu_{1}-\frac{\deg X}{r}\sum_{i=1}^{r}(i-1)\\& = \mu_{max}(\mathcal{F}|_{Y})-\deg X\cdot\frac{r-1}{2}. \end{aligned}

◻

Semistability and Tensor Products

One thing we need to prove in this section is tensor product preserves semistability. The proof involves Theorem 21 and ampleness of positive degree sheaves, which become true only in characteristic zero case. We need to first figure out the behaviour of semistable sheaves under pullback and pushforward via finite morphisms.

Let $f:Y\to X$ be a finite morphism of normal projective varieties over $k$ of dimension $n$ . Fix an ample line sheaf $\mathcal{O}_{X}(1)$ , then $\mathcal{O}_{Y}(1)=f^{*}\mathcal{O}_{X}(1)$ is also ample. $f$ is affine morphism so all the higher direct image vanishes, and $H^{i}(X,f_{*}\mathcal{F})=H^{i}(Y,\mathcal{F})$ . In particular, the Hilbert polynomial $P(\mathcal{F})(m)=P(f_{*}\mathcal{F})(m)$ and therefore $f_{*}$ preserves the dimension of sheaves.

Let $\mathcal{A}$ be the sheaf of algebras $f_{*}\mathcal{O}_{Y}$ , then $\mathcal{A}$ is a torsion-free coherent sheaf of rank $d$ . $f_{*}$ gives an equivalence between the category of coherent sheaves on $Y$ and the category of coherent sheaves on $X$ with $\mathcal{A}$ -module structure. $\mathcal{A}$ is torsion-free and thus locally free in codimension 1, which means $f$ is flat in codimension 1. $f^{*}$ is exact functor from the quotient category $\mathrm{Coh}_{n,n-1}(X)$ to $\mathrm{Coh}_{n,n-1}(Y)$ .

For a pure sheaf $\mathcal{F}$ of dimension $m$ , suppose there’s a $n$ -dimensional quotient $\mathcal{G}$ of $f_{*}\mathcal{F}$ . $\mathcal{G}$ admits a natural $\mathcal{A}$ -module structure, so there’s a coherent $\mathcal{O}_{Y}$ -module $\mathcal{G}'$ , which is a $m$ -dimensional quotient of $\mathcal{F}$ . This contradicts the assumption that $\mathcal{F}$ is pure. Therefore $f_{*}$ preserves the purity of sheaves.

Assume $\mathcal{F}$ is pure of dimension $n$ which is torsion-free in codimension 1 in $\mathrm{Coh}(X)$ . Since $\deg(f^{*}\mathcal{F})=d\deg(\mathcal{F})$ and $\mathop{\mathrm{rk}}(f^{*}\mathcal{F})=\mathop{\mathrm{rk}}(\mathcal{F})$ , $\mu(f^{*}\mathcal{F})=d\mu(\mathcal{F})$ .

Assume $\mathcal{G}$ is pure of dimension $n$ which is torsion-free in codimension 1 in $\mathrm{Coh}(Y)$ . Then $c_{1}(\mathcal{O}_{Y}(1))^{n}=d\cdot c_{1}(\mathcal{O}_{X}(1))^{n}$ and $\deg Y=d\deg X$ . Since $P(\mathcal{G})(m)=P(f_{*}\mathcal{G})(m)$ , we have $\mathop{\mathrm{rk}}(f_{*}\mathcal{G})=d\mathop{\mathrm{rk}}(\mathcal{G})$ . Note that

\deg(\mathcal{A})=\alpha_{d-1}(\mathcal{O}_{Y})-\mathop{\mathrm{rk}}(\mathcal{A})\alpha_{d-1}(\mathcal{O}_{X})=\alpha_{d-1}(\mathcal{O}_{Y})-d\cdot \alpha_{d-1}\mathcal{O}_{X},

\deg(\mathcal{G})=\alpha_{d-1}(\mathcal{G})-\mathop{\mathrm{rk}}(\mathcal{G})\alpha_{d-1}(\mathcal{O}_{Y})=\alpha_{d-1}(f_{*}\mathcal{G})-\mathop{\mathrm{rk}}(f_{*}\mathcal{G})\alpha_{d-1}(\mathcal{O}_{X})-\mathop{\mathrm{rk}}(\mathcal{G})\deg(\mathcal{A}).

We have $\mu(\mathcal{G})=d(\mu(f_{*}\mathcal{G})-\mu(\mathcal{A}))$ .

In characteristic zero case, we have the following lemma:

Lemma 23. Let $\mathcal{F}$ be a $n$ -dimensional coherent sheaf on $X$ . Then $\mathcal{F}$ is semistable if and only if $f^{*}\mathcal{F}$ is semistable.

Proof

Proof. $\mathcal{F}$ is torsion-free in codimension 1 if and only if $f^{*}\mathcal{F}$ is torsion-free, so we are allowed to work in the category $\mathrm{Coh}_{n,n-1}(Y)$ .

We first show the if direction: Suppose there is subsheaf $\mathcal{E}$ of $\mathcal{F}$ such that $\mu(\mathcal{E})>\mu(\mathcal{F})$ , then $\mu(f^{*}\mathcal{E})>\mu(\mathcal{F})$ . This leads to a contradiction.

Then we show the only if direction: Let $K$ be the splitting field of the function field $K(Y)$ over $K(X)$ . Let $Z$ be the normalization of $Y$ in $K$ , then we have finite morphisms $Z\to Y\to X$ . By pulling back $f^{*}\mathcal{F}$ to $Z$ , we may consider the finite morphism $g:Z\to X$ . Note that $K(Z)$ is Galois over $K(X)$ , $Z\to X$ is a Galois cover with Galois group $G$ . Suppose $g^{*}\mathcal{F}$ is not semistable and it’s maximal destablizing sheaf $\mathcal{E}$ . Since $\mathcal{E}$ is unique, it’s invariant under the action of $G$ . By Theorem 43, there’s a coherent subsheaf $\mathcal{F}'\subset \mathcal{F}$ such that $f^{*}\mathcal{F}'$ isomorphic to $\mathcal{E}$ in codimension 1. Then $\mu(\mathcal{F}')>\mu(\mathcal{F})$ and we have a contradiction. ◻

We will omit the proof of the next lemma here and refer to Ma¹¹.

Lemma 24. Let $X$ be a projective normal variety over algebraically closed field $k$ and let $\mathcal{O}_{X}(1)$ be a very ample line sheaf. For integer $d$ there exist a projective normal variety $X'$ with very ample line sheaf $\mathcal{O}_{X'}(1)$ and a finite morphism $f:X'\to X$ such that $f^{*}\mathcal{O}_{X}(1)\cong \mathcal{O}_{X'}(d)$ . Moreover, if $X$ is smooth, $X'$ can be chosen to be smooth.

Proof

Proof. See Corollary 1.15.1 in Ma¹¹. ◻

Using above lemmas and results in Appendix B, we are able to prove the theorem:

Theorem 25. Let $X$ be a normal projective variety over an algebraically closed field of characteristic zero. If $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are semistable sheaves, then $\mathcal{F}_{1}\otimes \mathcal{F}_{2}$ is semistable.

Proof

Proof. Suppose on the contrary. Let $\mathcal{E}$ be a torsion-free destablizing quotient of $\mathcal{F}_{1}\otimes \mathcal{F}_{2}$ . We first reduce to the case that $\mu(\mathcal{F}_{1})+\mu(\mathcal{F}_{2})-\mu(\mathcal{E})$ is large enough case. Using Lemma 24, there’s a finite morphism $f:X'\to X$ such that $f^{*}\mathcal{O}_{X}(1)\cong \mathcal{O}_{X'}(d)$ for large $d$ . $\mu(f^{*}\mathcal{E})$ are defined with respect to $\mathcal{O}_{X'}(1)$ . $f^{*}\mathcal{F}_{1}$ and $f^{*}\mathcal{F}_{2}$ are also semistable by Lemma 23, and $f^{*}\mathcal{E}$ is a torsion free destablizing quotient of $f^{*}(\mathcal{F}_{1}\otimes \mathcal{F}_{2})$ . Take

d>\lceil{\frac{\deg X\cdot (\mathop{\mathrm{rk}}(\mathcal{F}_{1})+\mathop{\mathrm{rk}}(\mathcal{F}_{2})+2)}{2(\mu(\mathcal{F}_{1})+\mu(\mathcal{F}_{2})-\mu(\mathcal{E}))}}\rceil,

we have

\begin{aligned} \frac{\mu(f^{*}\mathcal{F}_{1})+\mu(f^{*}\mathcal{F}_{2})-\mu(f^{*}\mathcal{E})}{\deg X'}&=d\cdot \frac{\mu(\mathcal{F}_{1})+\mu(\mathcal{F}_{2})-\mu(\mathcal{E})}{\deg X}\\ &> \frac{\mathop{\mathrm{rk}}(\mathcal{F}_{1})+\mathop{\mathrm{rk}}(\mathcal{F}_{2})+2}{2}\\ &= \frac{\mathop{\mathrm{rk}}(f^{*}\mathcal{F}_{1})+\mathop{\mathrm{rk}}(f^{*}\mathcal{F}_{2})-2}{2}.\end{aligned}

So we may assume

\mu(\mathcal{F}_{1})+\mu(\mathcal{F}_{2})-\mu(\mathcal{E})>\frac{\deg X\cdot (\mathop{\mathrm{rk}}(\mathcal{F}_{1})+\mathop{\mathrm{rk}}(\mathcal{F}_{2})+2)}{2}.

According to Bertini theorem, the complete intersection for general $\dim X-1$ hyperplanes forms a smooth curve $C$ . Applying Corollary 22, the Harder-Narasimhan factors satisfy

\mu(gr_{j}^{HN}(\mathcal{F}_{i}|_{C}))\geq \mu(\mathcal{F}_{i})-\deg X\cdot \frac{\mathop{\mathrm{rk}}(\mathcal{F}_{i})-1}{2}

for each $i=1, 2$ . Let

n_{i}=\lceil{\frac{\mu(\mathcal{F}_{i})}{\deg X}-\frac{\mathop{\mathrm{rk}}(\mathcal{F}_{i})-1}{2}}\rceil-1.

Then

\mu(gr_{j}^{HN}\mathcal{F}_{i}(-n_{i})|_{C})\geq \mu(\mathcal{F}_{i})-\deg X\cdot (n_{i}+\frac{\mathop{\mathrm{rk}}(\mathcal{F}_{i})-1}{2})>0.

Then $gr_{j}^{HN}(\mathcal{F}_{i}(-n_{i})|_{C})$ is a semistable torsion-free sheaf on $C$ and therefore a semistable vector bundle by 45. $\mu(gr_{j}^{HN}(\mathcal{F}_{i}(-n_{i})|_{C}))>0$ so by Theorem 50 $gr_{j}^{HN}(\mathcal{F}_{i}(-n_{i})|_{C})$ is ample. According to Corollary 49, $gr_{i}^{HN}(\mathcal{F}_{1}(-n_{1})|_{C})\otimes gr_{j}^{HN}(\mathcal{F}_{2}(-n_{2})|_{C})$ is ample. Therefore by Proposition 47 $(\mathcal{F}_{1}\otimes \mathcal{F}_{2})(-n_{1}-n_{2})|_{C}$ is also ample. $\mathcal{E}(-n_{1}-n_{2})|_{C}$ as a quotient of ample vector bundle is also ample.

Note that

\begin{aligned} \mu(\mathcal{E}(-n_{1}-n_{2}))&=\mu(\mathcal{E})-(n_{1}+n_{2})\deg X\\ &<\mu(\mathcal{F}_{1})+\mu(\mathcal{F}_{2})-\deg X\cdot (n_{1}+n_{2}+\frac{\mathop{\mathrm{rk}}(\mathcal{F}_{1})+\mathop{\mathrm{rk}}(\mathcal{F}_{2})+2}{2})\\ &=\mu(\mathcal{F}_{1})-\deg X\cdot (\frac{\mathop{\mathrm{rk}}(\mathcal{F}_{1})-1}{2}+n_{1}+1)\\ &+\mu(\mathcal{F}_{2})-\deg X\cdot (\frac{\mathop{\mathrm{rk}}(\mathcal{F}_{2})-1}{2}+n_{2}+1)\\ &\leq 0.\end{aligned}

So $\deg(\mathcal{E}(-n_{1}-n_{2})|_{C})=\deg(\mathcal{E}(-n_{1}-n_{2}))<0$ , which contardicts the fact $\mathcal{E}(-n_{1}-n_{2})|_{C}$ is ample. ◻

Corollary 26. Let $\mathcal{F}$ and $\mathcal{G}$ be torsion-free coherent sheaves on normal projective variety $X$ . Then

$\mu_{min}(\mathcal{F}\otimes\mathcal{G})=\mu_{min}(\mathcal{F})+\mu_{max}(\mathcal{G})$ .
$\mu_{max}(\mathcal{F}\otimes\mathcal{G})=\mu_{max}(\mathcal{F})+\mu_{max}(\mathcal{G})$ .

Proof

Proof. We only prove for $\mu_{max}$ . One direction is easy: As in Remark 11, pick the maximal destablizing sheaf $\mathcal{F}'$ and $\mathcal{G}'$ of $\mathcal{F}$ , $\mathcal{G}$ respectively. Then $\mathcal{F}'\otimes\mathcal{G}'$ destablizes $\mathcal{F}\otimes\mathcal{G}$ and $\mu(\mathcal{F}')+\mu(\mathcal{G}')=\mu_{max}(\mathcal{F})+\mu_{max}(\mathcal{G})$ . So $\mu_{max}(\mathcal{F}\otimes\mathcal{G})\geq \mu_{max}(\mathcal{F})+\mu_{max}(\mathcal{G})$ .

For the other direction, $\mathcal{F}$ and $\mathcal{G}$ are locally free in codimension 1, we may work in the cateogory $\mathrm{Coh}_{\dim X,\dim X-1}(X)$ . First we prove for the case $\mathcal{G}$ is semistable. Let $gr_{i}^{HN}(\mathcal{F})=HN_{i}(\mathcal{F}) /HN_{i-1}(\mathcal{F})$ be the Harder-Narasimhan filtration of $\mathcal{F}$ . Then

0=HN_{0}\mathcal{F}\otimes\mathcal{G}\subset \cdots\subset HN_{n}(\mathcal{F})\otimes \mathcal{G}=\mathcal{F}\otimes \mathcal{G}

gives a filtration of $\mathcal{F}\otimes \mathcal{G}$ with semistable factors and strictly decreasing slope. Thus the uniqueness of Harder-Narasimhan filtration shows that $gr_{i}^{HN}(\mathcal{F})\otimes \mathcal{G}$ gives a Harder-Narasimhan filtration of $\mathcal{F}\otimes\mathcal{G}$ . $HN_{0}(\mathcal{F})\otimes\mathcal{G}$ is the maximal destablizing sheaf of $\mathcal{F}\otimes\mathcal{G}$ and $\mu_{max}(\mathcal{F}\otimes\mathcal{G})=\mu_{max}(\mathcal{F})+\mu_{max}(\mathcal{G})$ .

For general $\mathcal{G}$ , we prove $\mu_{max}(gr_{i}^{HN}(\mathcal{F})\otimes\mathcal{G})=\mu_{max}(\mathcal{F})+\mu_{max}(\mathcal{G})$ by induction on $i$ . The base case is from our discussion above. For $i>1$ , consider the exact sequence

0\to HN_{i-1}(\mathcal{F})\otimes\mathcal{G}\to HN_{i}(\mathcal{F})\otimes\mathcal{G}\to gr_{i}^{HN}(\mathcal{F})\otimes\mathcal{G}\to 0

in the category $\mathrm{Coh}_{\dim X,\dim X-1}$ . Then by Remark 6,

\mu_{max}(HN_{i-1}(\mathcal{F})\otimes\mathcal{G})\leq \mu_{max}(HN_{i}(\mathcal{F})\otimes\mathcal{G}).

Let $\mathcal{E}$ be the maximal destablizing sheaf of $HN_{i}(\mathcal{F})\otimes\mathcal{G}$ . Let $0\to \mathcal{E}'\to \mathcal{E}\to \mathcal{E}' /\mathcal{E}\to 0$ be the induced exact sequence by setting $\mathcal{E}'=\mathcal{E}\cap HN_{i-1}(\mathcal{F})\otimes\mathcal{G}$ , then $\mathcal{E}' /\mathcal{E}$ is a subsheaf of $gr_{i}^{HN}(\mathcal{F})\otimes\mathcal{G}$ . So we have

\begin{aligned} \mu_{max}(HN_{i}(\mathcal{F}\otimes\mathcal{G}))&=\mu(\mathcal{E}) \leq \max\{\mu(\mathcal{E}),\mu(\mathcal{E}' /\mathcal{E})\}\\ &\leq \max\{\mu_{max}(HN_{i-1}(\mathcal{F})\otimes\mathcal{G}),\mu_{max}(gr_{i}^{HN}(\mathcal{F})\otimes\mathcal{G})\}\\ &=\mu_{max}(HN_{i-1}(\mathcal{F})\otimes\mathcal{G})\end{aligned}

Thus $\mu_{max}(HN_{i}(\mathcal{F})\otimes\mathcal{G})=\mu_{max}(\mathcal{F})+\mu_{max}(\mathcal{G})$ . ◻

Corollary 27. Let $\mathcal{F}$ be a torsion-free semistable sheaf on $X$ . Then all exterior products $\wedge^{n}\mathcal{F}$ , all symmetric products $S^{n}\mathcal{F}$ and $\mathop{\mathcal{H}om}(\mathcal{F},\mathcal{F})$ are semistable.

Proof

Proof. Let $\dim X=d$ . Note that $\wedge^{n}\mathcal{F}$ and $S^{n}\mathcal{F}$ are all direct summands of $\mathcal{F}^{\otimes n}$ . On can easily calculate that $\mu(\mathcal{F}^{\otimes n})=\mu(\wedge^{n}\mathcal{F})=\mu(S^{n}\mathcal{F})$ , so according to Remark 6, they are semistable.

We show that $\check{\mathcal{F}}$ is semistable. Suppose there is a destablizing sheaf $\mathcal{E}$ of $\check{\mathcal{F}}$ . We may assume $\check{\mathcal{F}} /\mathcal{E}$ is torsion-free, otherwise we may replace $\mathcal{E}$ by its saturation in $\check{\mathcal{F}}$ . Therefore $\check{\mathcal{F}} /\mathcal{E}$ is locally free in codimension 1. Consider the exact sequence

0\to \mathcal{E}\to \check{\mathcal{F}}\to \check{\mathcal{F}} /\mathcal{E}\to 0

in $\mathrm{Coh}_{d,d-1}(X)$ , it’s dual

0\to (\check{\mathcal{F}} /\mathcal{E})^{\check{}}\to \mathcal{F}\to \check{\mathcal{E}}\to 0

in $\mathrm{Coh}_{d,d-1}(X)$ is still exact. Then $\mu(\mathcal{E})>\mu(\check{\mathcal{F}})$ implies $\mu(\mathcal{F})>\mu(\check{\mathcal{E}})$ , which contradict the assumption $\mathcal{F}$ is semistable. In codimension 1 $\mathcal{F}$ is locally free so $\mathop{\mathcal{H}om}(\mathcal{F},\mathcal{F})$ is isomorphic to $\mathcal{F}\otimes\check{\mathcal{F}}$ in $\mathrm{Coh}_{d,d-1}(X)$ . Since $\mathcal{F}$ and $\check{\mathcal{F}}$ are semistable, $\mathop{\mathcal{H}om}(\mathcal{F},\mathcal{F})$ is semistable. ◻

Boundedness in Zero Characteristic

In this section, we will use Theorem 17 to show the boundedness of semistable sheaves with fixed Hilbert polynomial. The base field $k$ is assumed to be algebraically closed with characteristic zero. Let $[x]_{+}=\max\{x,0\}$ for any real number $x$ .

We first deal with the case that $X$ is normal.

Lemma 28. Let $X$ be a normal projective variety of dimension $d$ . Let $\mathcal{F}$ be a torsion-free sheaf on $X$ . Then for any $\mathcal{F}$ -regular sequence of hyperplane sections $H_{1},\dots, H_{d}$ and $X_{v}=H_{1}\cap \cdots\cap H_{d-v}$ the following inequality holds for all $v=1,\dots, d$ :

\frac{h^{0}(X_{v},\mathcal{F}|_{X_{v}})}{\mathop{\mathrm{rk}}(\mathcal{F})\cdot \deg X}\leq \frac{1}{v!}\left[\frac{\mu_{max}(\mathcal{F}|_{X_{1}})}{\deg X}+v\right]_{+}^{v}.

Proof

Proof. The proof proceeds by induction on $v$ . For the base case, let $gr_{i}^{HN}(\mathcal{F}|_{X_{1}})$ , $i=1,\dots, l$ be the Harder-Narasimhan filtration of $\mathcal{F}|_{X_{1}}$ . Taking the global sections one have

h^{0}(X_{1},\mathcal{F}|_{X_{1}})\leq \sum_{i=1}^{l}h^{0}(X_{1},gr_{i}^{HN}(\mathcal{F}|_{X_{1}})).

We may assume $\mathcal{F}|_{X_{1}}$ is semistable and therefore $\mu_{max}(\mathcal{F}|_{X_{1}})=\mu(\mathcal{F}|_{X_{1}})$ . For any $n\geq 0$ , we can take $\mathcal{F}|_{X_{1}}$ -regular section $H$ in $\mathcal{O}_{X_{1}}(n)$ and an exact sequence

0\to \mathcal{F}|_{X_{1}}(-n)\to \mathcal{F}|_{X_{1}}\to \mathcal{F}|_{X_{1}\cap H}\to 0

So

h^{0}(X_{1},\mathcal{F}|_{X_{1}})\leq h^{0}(X_{1},\mathcal{F}|_{X_{1}})+n\cdot\mathop{\mathrm{rk}}(\mathcal{F})\cdot \deg X

For $n\geq \lceil{\frac{\mu(\mathcal{F}|_{X_{1}})}{\deg X}}\rceil+1$ , we have $h^{0}(X_{1},\mathcal{F}|_{X_{1}}(-n))=\mathop{\mathrm{hom}}(\mathcal{O}_{X_{1}}(n),\mathcal{F}|_{X_{1}})=0$ . Thus we get our desired upper bound.

Assume the inequality for $v-1$ with $v\geq 2$ . Consider the exact
sequence

0\to \mathcal{F}|_{X_{v}}(-k-1)\to \mathcal{F}|_{X_{v}}(-k)\to \mathcal{F}|_{X_{v-1}}(-k)\to 0

for $k=0,1,2,\dots$ , inductively we have

\begin{aligned} h^{0}(X_{v},\mathcal{F}|_{X_{v}})&\leq h^{0}(X_{v},\mathcal{F}|_{X_{v}}(-n))+\sum_{k=0}^{n-1}h^{0}(X_{v-1},\mathcal{F}|_{X_{v-1}}(-k))\\ &\leq \sum_{k=0}^{\infty}h^{0}(X_{v-1},\mathcal{F}|_{X_{v-1}}(-k)).\end{aligned}

Similarly, $h^{0}(X_{v-1},\mathcal{F}|_{X_{v-1}}(-k))$ vanishes for $k>\mu_{max}(\mathcal{F}|_{X_{v-1}})$ so the summation is actually finite. Using the induction hypothesis and replace the summation by integral, we have

\frac{h^{0}(X_{v},\mathcal{F}|_{X_{v}})}{\mathop{\mathrm{rk}}(\mathcal{F})\cdot \deg X}\leq \frac{1}{(v-1)!}\int_{-1}^{C}\left[\frac{\mu_{max}(\mathcal{F}_{X_{1}})}{\deg X}+v-1-t\right]_{+}^{v-1}dt

where $C$ is the maximum of -1 and the smallest zero of the integrand. By simple calculus we have the right hand side of the lemma. ◻

Using above lemma and Corollary 22, we immediately have

Corollary 29. Let $X$ be a normal projective variety of dimension $d$ and $\mathcal{F}$ is torsion-free sheaf on $X$ . For general hyperplanes $H_{1},\dots, H_{d}$ in $\left| \mathcal{O}_{X}(1) \right|$ , the following inequality holdes for all $v=1,\dots, d$ :

\frac{h^{0}(X_{v},\mathcal{F}|_{X_{v}})}{\mathop{\mathrm{rk}}(\mathcal{F})\cdot \deg X}\leq \frac{1}{v!}\left[\frac{\mu_{max}(\mathcal{F})}{\deg X}+\frac{\mathop{\mathrm{rk}}(\mathcal{F})-1}{2}+v\right]_{+}^{v}.

Corollary 29 gives a uniform bound for torsion-free semistable sheaves with fixed Hilbert polynomial (note $\mu_{max}(\mathcal{F})=\mu(\mathcal{F})$ ) on normal projective variety. Combining it with 17, we get the boundedness for the specific case.

Theorem 30 (Le Potier-Simpson). Let $X$ be a $d$ -dimensional projective variety and $\mathcal{F}$ be a torsion free sheaf on $X$ . Let $r(\mathcal{F})=\alpha_{d}(\mathcal{F})$ be the multiplicity of $\mathcal{F}$ . Then there is a $\mathcal{F}$ -regular sequence of hyperplane sections $H_{1},\dots, H_{d}$ and $X_{v}=H_{1}\cap \cdots\cap H_{d-v}$ such that the following inequality holds for all $v=1,\dots, d$ :

\frac{h^{0}(X_{v},\mathcal{F}|_{X_{v}})}{r(\mathcal{F})}\leq \frac{1}{v!}\left[\hat{\mu}_{max}(\mathcal{F})+r(\mathcal{F})^{2}+\frac{r(\mathcal{F})+d}{2}-1\right]_{+}^{v}

Proof

Proof. Let $i:X\to \mathbb{P}^{N}$ be the closed immersion corresponding to the very ample sheaf $\mathcal{O}_{X}(1)$ . Consider a $N-d-1$ -dimensional linear subspace $L\subset \mathbb{P}^{N}$ which does not intersect $X$ . Let $\pi:\mathbb{P}^{N}-L\to Y\cong \mathbb{P}^{d}$ be the projection with center $L$ , $\pi$ is a finite morphism. Denote $\pi_{*}\mathcal{O}_{X}$ by $\mathcal{A}$ . $\pi_{*}\mathcal{F}$ is also torsion-free and $r(\mathcal{F})=\alpha_{d}(\pi_{*}\mathcal{F})=\mathop{\mathrm{rk}}(\pi_{*}\mathcal{F})$ . We also have

\hat{\mu}(\mathcal{F})=\hat{\mu}(\pi_{*}\mathcal{F})=\frac{\mu(\pi_{*}\mathcal{F})}{\alpha_{d}(\mathcal{O}_{Y})}+\hat{\mu}(\mathcal{O}_{Y})=\mu(\pi_{*}\mathcal{F})+\frac{d+1}{2}.

A $\pi_{*}\mathcal{F}$ -regular sequence $H_{i}'$ in $Y\cong \mathbb{P}^{d}$ naturally induces an $\mathcal{F}$ -regular sequence $H_{i}$ in $X$ . Denote $Y_{v}=H_{1}'\cap \cdots\cap H_{v}'$ , then $\pi_{*}(\mathcal{F}|_{X_{v}})=(\pi_{*}\mathcal{F})|_{Y_{v}}$ . Apply Corollary 29 to $\pi_{*}\mathcal{F}$ , there is an inequality for $v=1,\dots,d$ :

\frac{h^{0}(X_{v},\mathcal{F}|_{X_{v}})}{\mathop{\mathrm{rk}}(\pi_{*}\mathcal{F})}\leq \frac{1}{v!}\left[\mu_{max}(\pi_{*}\mathcal{F})+\frac{\mathop{\mathrm{rk}}(\pi_{*}\mathcal{F})-1}{2}+v\right]_{+}^{v}.

We need to estimate $\mu_{max}(\pi_{*}\mathcal{F})$ by $\hat{\mu}_{max}(\pi_{*}\mathcal{F})$ . First we show that $\mu_{min}(\mathcal{A})\geq -r(\mathcal{F})^{2}$ : Clearly $\mathcal{A}$ is torsion-free sheaf. The injection $\mathcal{A}\to \mathop{\mathcal{H}om}(\pi_{*}\mathcal{F},\pi_{*}\mathcal{F})$ shows that $\mathop{\mathrm{rk}}(\mathcal{A})\leq \mathop{\mathrm{rk}}(\pi_{*}\mathcal{F})^{2}=r(\mathcal{F})^{2}$ . Let $\mathcal{W}=\mathcal{O}_{Y}(-1)^{N-d}$ , then $X$ is a closed subscheme of the vector bundle $\pi: \mathbb{P}^{N}-L\cong \mathop{\mathrm{Spec}}S^{*}W\to Y$ , so there is a surjection $\varphi:S^{*}\mathcal{W}\to \mathcal{A}$ . Consider the filtration of $\mathcal{A}$ by ascending $\mathcal{O}_{Y}$ -modules

F_{i}\mathcal{A}=\varphi(\mathcal{O}_{Y}\oplus\mathcal{W}\oplus\cdots\oplus S^{i}\mathcal{W})

Since $\mathcal{A}$ is coherent, only finitely many factors $gr_{i}^{F}(\mathcal{A})$ are not zero. Since $\mathcal{W}\otimes gr_{i}^{F}(\mathcal{A})\to gr_{i+1}^{F}(\mathcal{A})$ is surjective, once $gr_{i}^{F}(\mathcal{A})$ has torsion, all $gr_{j}^{F}(\mathcal{A})$ has torsion for $j\geq i$ . If $gr_{i}^{F}(\mathcal{A})$ has no torsion then $i\leq \mathop{\mathrm{rk}}(\mathcal{A})$ , so the cokernel of $\varphi:\mathcal{O}_{Y}\oplus \mathcal{W}\oplus\cdots\oplus S^{\mathop{\mathrm{rk}}(\mathcal{A})}\mathcal{W}\to \mathcal{A}$ has torsion. Hence

\mu_{min}(\mathcal{A})\geq \mu_{min}(S^{\mathop{\mathrm{rk}}(\mathcal{A})}\mathcal{W})=\mu(S^{\mathop{\mathrm{rk}}(\mathcal{A})\mathcal{W}})=\mathop{\mathrm{rk}}(\mathcal{A})\mu(\mathcal{W})=-rk(\mathcal{A})

and $$\mu_{min}(\mathcal{A})\geq -r(\mathcal{F})^{2}.$$

Let $\mathcal{E}$ be the maximal destablizing sheaf of $\pi_{*}\mathcal{F}$ and $\mathcal{E}'$ be its image under the multiplication morphism $\mathcal{A}\otimes\mathcal{E}\to \mathcal{A}\otimes \pi_{*}\mathcal{F}\to \pi_{*}\mathcal{E}$ . $\mathcal{E}'$ is the $\mathcal{A}$ -submodule of $\pi_{*}\mathcal{F}$ generated by $\mathcal{E}$ , and $\mathcal{E}'\cong \pi_{*}\mathcal{E}''$ for some $\mathcal{O}_{X}$ -submodule $\mathcal{E}''$ of $\mathcal{F}$ . So

\hat{\mu}_{max}(\mathcal{F})\geq \hat{\mu}(\mathcal{E}'')=\hat{\mu}(\mathcal{E}')=\mu(\mathcal{E}')+\hat{\mu}(\mathcal{O}_{Y}),

by Lemma 5,

\mu(\mathcal{E}')+\hat{\mu}(\mathcal{O}_{Y})\geq \mu_{min}(\mathcal{A}\otimes\mathcal{E})+\frac{d+1}{2}

by Corollary 26,

\begin{aligned} \mu_{min}(\mathcal{A}\otimes\mathcal{E})+\frac{d+1}{2}&=\mu(\mathcal{E})+\mu_{min}(\mathcal{A})+\frac{d+1}{2}\\ &\geq \mu_{max}(\pi_{*}\mathcal{F})-r(\mathcal{F})^{2}+\frac{d+1}{2}.\end{aligned}

Therefore we have

\mu_{max}(\pi_{*}\mathcal{F})+\frac{\mathop{\mathrm{rk}}(\pi_{*}\mathcal{F})-1}{2}+v \leq \hat{\mu}_{max}(\mathcal{F})+r(\mathcal{F})^{2}+\frac{r(\mathcal{F})-1}{2}+\frac{d-1}{2}.

◻

Theorem 31. Let $X$ be a projective variety over $k$ . The family of semistable sheaves on $X$ with fixed Hilbert polynomial is bounded.

Proof

Proof. Immediate corollary from Theorem 17 and Theorem 30. ◻

Boundedness in General Characteristic

When $X$ is a smooth projective curve over algebraically closed field $k$ , the semistability of sheaves behave well in arbitrary characteristic.

Theorem 32. $\{\mathcal{F}_{i}\}$ is a family of semistable sheaves with fixed Hilbert polynomial, then $\{\mathcal{F}_{i}\}$ is bounded.

Proof

Proof. We show that there is an integer $m$ such that $\mathcal{F}_{i}$ is $m$ -regular for all $i$ . Assume the Hilbert polynomial is given by $P(n)=n\cdot\mathop{\mathrm{rk}}(\mathcal{F}_{i})\cdot\deg(\mathcal{O}(1))+\deg(\mathcal{F}_{i})+\mathop{\mathrm{rk}}(\mathcal{F}_{i})(1-g)$ , so all the sheaves have the same rank and degree. By Serre duality, $H^{1}(X,\mathcal{F}_{i}(m-1))=\mathop{\mathrm{Hom}}(\mathcal{F}_{i}(m-1),\omega)^{\check{}}$ . Note $\mathcal{F}_{i}(m-1)$ and $\omega$ are all semistable. Set

m=\lceil{\frac{2\mathop{\mathrm{rk}}(\mathcal{F}_{i})g(X)-2\mathop{\mathrm{rk}}(\mathcal{F}_{i})-\deg(\mathcal{F}_{i})}{\deg(\mathcal{O}(1))}+1}\rceil

and using Lemma 5, $\mathop{\mathrm{Hom}}(\mathcal{F}_{i}(m-1),\omega)=0$ for all $i$ . Thus $\{F_{i}\}$ is bound. ◻

Dimension 2 case is much more complicated since there is no good estimation for $H^{1}(X,\mathcal{F})$ . We will show that the family of semistable vector bundle with rank 2 on a smooth surface is bounded. Let $X$ be a smooth projective surface and $\mathcal{O}(1)$ be a very ample line sheaf. Let $\mathfrak{F}(P)$ be the family consists of rank 2 torsion-free semistable sheaf with fixed Hilbert polynomial $P$ .

Lemma 33 (Ta¹³). There are integer $n_{1}$ and $n_{2}$ such that for any $\mathcal{F}\in \mathfrak{F}(P)$ , there is a subsheaf $\mathcal{E}$ of $\mathcal{F}$ with $n_{1}\leq \deg(\mathcal{E})\leq n_{2}$ .

Proof

Proof. First $\mathcal{F}$ is semistable so one can set $n_{2}=\frac{\deg (\mathcal{F})}{2}$ , then for all rank 1 subsheaf $\mathcal{E}$ we have $\deg(\mathcal{E})\leq \frac{\deg (\mathcal{F})}{2}$ .

We can choose $n_{1}>0$ such that $P(\mathcal{F})(n)>0$ for all $\mathcal{F}\in \mathfrak{F}(P)$ . We may also assume $\deg (\mathcal{F})\geq -2n_{1}+2\deg (\omega)$ so $\deg(\check{\mathcal{F}}(-m)\otimes\omega)<0$ . Note that any global section will induce a morphism $\mathcal{O}_{X}\to\check{\mathcal{F}}(-m)\otimes\omega$ , which contradicts the fact $\check{\mathcal{F}}(-m)\otimes\omega$ is semistable with negative degree. So

H^{0}(X,\check{\mathcal{F}}(-m)\otimes\omega)=\mathop{\mathrm{Hom}}(\mathcal{F}(m),\omega)=H^{2}(X,\mathcal{F}(m))^{\check{}}=0.

Thus $H^{0}(X,\mathcal{F}(m))\neq 0$ for all $\mathcal{F}$ and a global section induces the morphism $\mathcal{O}_{X}\to \mathcal{F}(m)$ . Let the image be $\mathcal{G}$ , then $\deg(\mathcal{G}(n_{1}))\geq 0$ and $\deg(\mathcal{G})\geq -n_{1}$ . ◻

Theorem 34. $\mathfrak{F}(P)$ is bounded.

Proof

Proof. We will use Theorem 17 to prove this theorem. From Lemma 15 and the definition of Castelnuovo-Mumford regularity, if the family $\{\mathcal{F}(m)|\mathcal{F}\in \mathfrak{F}(P)\}$ is bounded, then $\mathfrak{F}(P)$ is bounded. So we may let $m$ small enough and assume $\deg(\mathcal{F})<0$ for all $\mathcal{F}\in \mathfrak{F}(P)$ . Since any nonzero global section of $\mathcal{F}$ will induce a morphism $\mathcal{O}_{X}\to \mathcal{F}$ , which will contradict to the assumption $\mathcal{F}$ is semistable. Thus $H^{0}(X,\mathcal{F})=0$ for all $\mathcal{F}$ .

For general hyperplane $H_{1}$ , $H_{2}$ in the linear system $\left| \mathcal{O}(1) \right|$ , $h^{0}(\mathcal{F}|_{H_{1}\cap H_{2}})=2\deg X$ is bounded.

Now we show $h^{0}(\mathcal{F}|_{H})$ is bounded for general hyperplane $H$ . According to Lemma 33, there are constants $n_{1}$ , $n_{2}$ and a subsheaf $\mathcal{E}\subset \mathcal{F}$ such that $n_{1}\leq \deg(\mathcal{E})\leq n_{2}$ . We may assume $X\cap H$ is smooth curve and $\mathcal{E}|_{H}$ , $(\mathcal{F}/\mathcal{E})|_{H}$ are locally free. Let $g$ be the genus of $X\cap H$ for general $H$ . Let $n_{3}=\deg(\mathcal{F})-n_{1}$ , $n_{4}=\deg(\mathcal{F})-n_{2}$ and $n=\max\{0,2g-n_{1},2g-n_{4}\}$ . Then $n_{4}\leq \deg(\mathcal{F}/\mathcal{E})\leq n_{3}$ . Since $\deg(\mathcal{E}|_{H})>2g-2$ and $\deg ((\mathcal{F}/\mathcal{E})|_{H})>2g-2$ , $h^{1}(\mathcal{E}|_{H})=0$ and $h^{1}((\mathcal{F}/\mathcal{E})|_{H})=0$ . By Riemann-Roch theorem,

h^{0}(\mathcal{E}|_{H}(n))\leq n+1-g+n_{2},

h^{0}((\mathcal{F}/\mathcal{E})|_{H}(n))\leq n+1-g+\deg(\mathcal{F})-n_{1}.

So

\begin{aligned} h^{0}(\mathcal{F}|_{H})&\leq h^{0}(\mathcal{E}|_{H})+h^{0}((\mathcal{F}/\mathcal{E})|_{H})\\ &\leq h^{0}(\mathcal{E}|_{H}(n))+h^{0}((\mathcal{F}/\mathcal{E})|_{H}(n))\\ &\leq 2-2g+2n+n_{2}-n_{1}\end{aligned}

is bounded. ◻

In higher dimensional, the traditional approach for boundedness won’t work when the base field is of positive characteristic. Theorem 21, Lemma 23 and Theorem 50 strongly depend on the assumption of characteristic. The problem in positive characteristic case is completely solved by Langer in his recent work Langer⁸.

In fact, Gieseker provided a counter-example for Lemma 23 in positive characteristic in Ge²: Assume the base field has characteristic $p$ . Let $F:X\to X$ be the absolute Frobenius morphism, i.e. the morphism which is an identity on topological space and sends $x\mapsto x^{p}$ on $\mathcal{O}_{X}$ .

Example 35 (Ge²). Let $p$ be any positive integer and $g\geq 2$ , then there is a curve $X$ of genus $g$ over a field of characteristic $p$ , and a semistable bundle $\mathcal{E}$ of rank 2 on $X$ , such that $F^{*}\mathcal{E}$ is not semistable.

The argument using ampleness does not work in positive characteristic case either. To make result similar to Lemma 25, Langer introduced strongly semistable in Langer⁸.

Definition 36. A coherent sheaf $\mathcal{F}$ in characteristic $p$ is strongly semistable if $(F^{n})^{*}\mathcal{F}$ is semistable for all $n\geq 0$ .

Langer showed in terms of strongly semistable, the tensor product of strongly semistable sheaves is still strongly semistable. He also improved the Bogomolov inequality and restriction theorems to general characteristic case and used them to show the boundness for family of sheaves with upper bound in $\mu_{max}$ .

There are several types restriction theorem for general characteristic case. For example, Mehta and Ramanathan proved their restriction theorem:

Theorem 37 (Mehta, Ramanathan). Let $X$ be a smooth projective variety of dimension $n\geq 2$ and let $\mathcal{O}(1)$ be a very ample line sheaf. Let $\mathcal{F}$ be a semistable sheaf. Then there is an integer $a_{0}$ such that for all $a\geq a_{0}$ there is an open dense subset $U_{a}\subset \left| \mathcal{O}(a) \right|$ such that for all $D\in U_{a}$ the divisor $D$ is smooth and $E|_{D}$ is semistable.

However, we cannot directly apply this theorem to prove the boundedness. Maruyama showed a theorem that improves $a_{0}$ to 1 when the rank of sheaves is small in Ma¹⁰:

Theorem 38. Let $X$ be a smooth projective variety of dimension $n\geq 2$ and let $\mathcal{O}(1)$ be a very ample line sheaf. Let $\mathcal{F}$ be a torsion-free semistable sheaf on $X$ and $\mathop{\mathrm{rk}}(\mathcal{F})<\dim X$ . Then for general hyperplane section $H$ in $\left| \mathcal{O}(1) \right|$ , the restriction $\mathcal{F}|_{H}$ is semistable.

Maruyama proved boundedness of semistable bundles of rank 2 with fixed Hilbert polynomial on smooth projective varieties using the above theorem. See Ma¹⁰ for more details.

Results from Descent Theory

In this section we introduce some basic results from descent of quasicoherent sheaves. We need faithfully flat descent and Galois descent in the Theorem 21 and Lemma 23. The proof of theorems in this section will be omitted and we refer chapter 14 Grotz&We³ and Stacks[0238]¹ for details.

Let $p:S'\to S$ be a faithfully flat quasicompact morphism of schemes. Let $S''=S'\times_{S}S'$ and $S'''=S'\times_{S}S'\times_{S}S''$ and the projections be $p_{i}:S''\to S'$ and $p_{ij}:S'''\to S''$ .

Definition 39. Let $\mathcal{F}$ be a quasicoherent sheaf on $S'$ . A descent datum of $\mathcal{F}$ is a $\mathcal{O}_{S'}$ -module morphism $\varphi:p_{1}^{*}\mathcal{F}\xrightarrow{\simeq}p_{2}^{*}\mathcal{F}$ satisfying the cocycle condition

p_{23}^{*}\varphi\circ p_{12}^{*}\varphi=p_{13}^{*}\varphi,

i.e. $\varphi$ makes the following diagram commutes:

We can define the category $\mathrm{Qcoh}(S' /S)$ of quasicoherent sheaves on $S'$ with the descent datum. More precisely, the object of $\mathrm{Qcoh}(S' /S)$ is pair $(\mathcal{F},\varphi)$ the quasicoherent sheaves on $S'$ with descent datum $\varphi$ . The morphism from $(\mathcal{F},\varphi)$ to $(\mathcal{G},\psi)$ is a morphism of $\mathcal{O}_{S}'$ -modules $u:\mathcal{F}\to \mathcal{G}$ such that $p_{2}^{*}u\circ \varphi=\psi \circ p_{1}^{*}u$ .

Let $\Phi:\mathrm{Qcoh}(S)\to \mathrm{Qcoh}(S' /S)$ be a functor which maps $\mathcal{F}$ to $p^{*}\mathcal{F}$ and give rise to a canonical datum $\varphi:p_{1}^{*}(p^{*}\mathcal{F})\xrightarrow{\simeq}p_{2}^{*}(p^{*}\mathcal{F})$ .

Theorem 40. $\Phi:\mathrm{Qcoh}(S)\to \mathrm{Qcoh}(S' /S)$ defines an equivalence of categories.

Next we study a specific case of faithfully flat descent, using the action of a Galois group. Let $G$ be a finite group and $S$ a scheme. We can view $\coprod\limits_{g\in G}S$ as the constant group scheme over $S$ : It represents the functor that associate $S$ -scheme $T$ with the locally constant map $T\to G$ . Denote the group scheme $\coprod\limits_{g\in G}$ by $G_{S}$ . The multiplication $\mu:G_{S}\times_{S}G_{S}\to G_{S}$ is given by transportation of $T$ -points in each component. An action of $G_{S}$ on a $S$ -scheme $S'$ by $S$ -automorphism is a morphism $\sigma: G_{S}\times_{S} S'\to S'$ , $\sigma:(g,s')\mapsto gs'$ in $T$ -points.

Definition 41. Let $p_{2}:G_{S}\times_{S}S'\to S'$ be the projection on the section component. A $G_{S}$ -equinvariant structure on quasicoherent $\mathcal{O}_{S}$ -module $\mathcal{F}$ is an isomorphism $\varphi:\sigma^{*}\mathcal{F}\xrightarrow{\simeq} p_{2}^{*}\mathcal{F}$ satisfying the cocycle condition

p_{23}^{*}\varphi \circ (id_{G_{S}}\times \sigma)^{*}\varphi=(\mu\times id_{S'})\varphi.

Definition 42. A Galois covering with Galois group $G_{S}$ is a faithfully flat morphism $p:S\to S$ with a $G_{S}$ -action on $S'$ by $S$ -automorphism such that the morphism $G_{S}\times_{S}S'\to S'\times_{S} S'$ which on $T$ -points $(g,s')\mapsto (s',gs')$ is an isomorphism.

Moreover, if $f:Y\to X$ is a finite morphism of normal projective schemes with $K(Y)$ Galois over $K(X)$ , then $f$ is a Galois covering.

Let $S'\to S$ be a Galois covering. We can define the $G_{S}$ -equinvariant quasicoherent sheaf category $\mathrm{Qcoh}_{G}(S' /S)$ : The objects in $\mathrm{Qcoh}_{G}(S' /S)$ are pairs $(\mathcal{F},\varphi)$ of quasicoherent sheaves on $S'$ with $G_{S}$ -equinvariant structure. A morphism from $(\mathcal{F},\varphi)$ to $(\mathcal{G},\psi)$ are $\mathcal{O}_{S'}$ -morphism $u:\mathcal{F}\to \mathcal{G}$ such that $\psi\circ \sigma^{*}u=p_{2}^{*}u\circ \varphi$ .

Similarly, we can define a functor $\Phi:\mathrm{Qcoh}(S)\to \mathrm{Qcoh}_{G}(S' /S)$ which maps $\mathcal{F}$ to $p^{*}$ and gives rise to canonical $G_{S}$ -equinvariant structure $p_{2}^{*}p^{*}\mathcal{F}\xrightarrow{\simeq}\sigma^{*}p^{*}\mathcal{F}$ .

Theorem 43. $\Phi:\mathrm{Qcoh}(S)\to \mathrm{Qcoh}_{G}(S' /S)$ defines an equivalence of categories.

Ample Vector Bundles

We will summarize the important relation between ample sheaf and positive degree sheaf in this section. The original reference for this section is Ha⁵, and we will mainly refer Ha⁵ and La⁹ for the proofs.

Let $X$ be a smooth projective variety over algebraically closed field $k$ , $\mathcal{E}$ is a vector bundle on $X$ and $\pi:\mathbb{P}(\mathcal{E})\to X$ be the projective bundle associated to $\mathcal{E}$ . Denote the tautological line sheaf on $\mathbb{P}(\mathcal{E})$ by $\mathcal{O}_{\pi}(1)$ .

Definition 44. $\mathcal{E}$ is ample if $\mathcal{O}_{\pi}(1)$ is ample;

$\mathcal{E}$ is nef if $\mathcal{O}_{\pi}(1)$ is nef.

The following proposition is straight from definition.

Proposition 45. Let $\mathcal{E}$ be an ample (nef) vector bundle on $X$ , then for any quotient $\mathcal{F}$ of $\mathcal{E}$ , $\mathcal{F}$ is ample (nef).

Theorem 46. Let $\mathcal{E}$ be a vector bundle on $X$ , then the followings are equivalent:

$\mathcal{E}$ is ample.
For any coherent sheaf $\mathcal{F}$ on $X$ , there is an integer $m(\mathcal{F})$ such that $H^{i}(X,S^{m}\mathcal{E}\otimes \mathcal{F})=0$ for all $i>0$ and $m\geq m(\mathcal{F})$ .
For any coherent sheaf $\mathcal{F}$ on $X$ , there is an integer $n(\mathcal{F})$ such that $S^{n}\mathcal{E}\otimes\mathcal{F}$ is generated by global sections for all $n\geq n(\mathcal{F})$ .

Using above theorem, one can show that:

Proposition 47. Let $0\to \mathcal{E}'\to \mathcal{E}\to \mathcal{E}''\to 0$ be an exact sequence of vector bundles on $X$ . If $\mathcal{E}'$ and $\mathcal{E}''$ are ample, then so is $\mathcal{E}$ .

Proposition 48. Let $\mathcal{E}$ be a vector bundle on $X$ , then the followings are equivalent:

$\mathcal{E}$ is ample.
$S^{k}\mathcal{E}$ is ample for some $k\geq 1$ .
$S^{k}\mathcal{E}$ is ample for all $k\geq 1$ .

Corollary 49. Assume $\mathcal{E}$ and $\mathcal{F}$ are ample vector bundles on $X$ . Then $\mathcal{E}\otimes\mathcal{F}$ is ample.

All propositions above are still true on varieties over $k$ with positive characteristic.

Theorem 50. Assume $k$ is algebraically closed field with characteristic zero. Let $X$ be a projective curve over $k$ and $\mathcal{E}$ be a semistable vector bundle of over $X$ . Then

$\mathcal{E}$ is nef if and only if $\deg(\mathcal{E})\geq 0$ ;
$\mathcal{E}$ is ample if and only if $\deg(\mathcal{E})> 0$ .

The assumption $\mathop{\mathrm{char}}k=0$ is required in Theorem 50. In fact, Serre constructed a non-singular curve $X$ of genus 3 over a field of characteristic 3, and a vector bundle $\mathcal{E}$ of rank 2 with $\deg(\mathcal{E})=1$ , while all the quotient of $\mathcal{E}$ has positive degree but not ample. See La⁴ for the example.

Reference

[1]: The Stacks Project Authors. Stacks project, 2024.

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[5]: Robin Hartshorne. Ample subvarieties of algebraic varieties, volume 156. Springer, 2006.

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[10]: Masaki Maruyama. Boundedness of semi-stable sheaves of small ranks. Nagoya Mathematical Journal, 78:65–94, 1980.

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[12]: David Mumford. Lectures on Curves on an Algebraic Surface. (AM-59). Princeton University Press, 1966.

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