Cohomology bounds and Chern class inequalities for stable sheaves on a smooth projective variety

Abstract

We give effective upper bounds for dimensions of the $(n-1)$-th cohomology groups of $\mu$-semistable torsion-free sheaves on a smooth projective variety of dimension n defined over an algebraically closed fieled of characteristic zero. As a corollary to this result, we obtain bounds for the dimension of the moduli space of $\mu$-stable vector bundles. We also prove Bogomolov-Gieseker type inequalities for the fourth Chern classes $c_4(E)$ of $\mu$-semistable vector bundles E on a smooth projective fourfold.

1 Introduction

Let X be a smooth projective variety defined over an algebraically closed field of characteristic 0 and let H be an ample line bundle on X. The classical Bogomolov-Gieseker inequality states that $\Delta _H(E)=(2rc_2(E)-(r-1)c_1(E)^2)\cdot H^{n-2}\ge 0$ for any torsion-free sheaf E of rank r and Chern classes $c_i(E)$ on X which is $\mu$-semistable with respect to H. Recently, some conjectures for the third Chern character $\mathrm{ch}_3(E)$ of $\mu$-stable sheaves E on a threefold have been proposed ( [1, 2]). We gave explicit bounds for the cohomology groups for $\mu$-semistable sheaves E on a threefold and applied them to obtain inequalities for $\mathrm{ch}_3(E)$ in [6, 7]. On the other hand, it seems that no Bogomolov-Gieseker type inequality has been known for the top Chern class $c_n(E)$ for $\mu$-semistable sheaves E on a variety of dimension $n\ge 4$.

In this note we give an effective bound for the dimension of the cohomology group $\mathrm{Ext}^1(E,E_1)$ for $\mu$-semistable torsion-free sheaves E and $E_1$ on a smooth projective variety of dimension $n\ge 3$. As in [7], we prove this by reducing the problem to the three dimensional case using the restriction theorem due to A.Langer ( [4, 5]) and a vanishing theorem of H.Sun ( [8]). As a corollary, we obtain upper bounds for the dimension of the moduli of $\mu$-stable vector bundles. We also obtain explicit upper bounds for $c_4(E)$ of $\mu$-semistable bundles E in terms of $r,\,c_i(E)$ $(1\le i\le 3)$, $\,c_i(X)$ and H on a smooth projective fourfold.

2 Notations and Preliminaries

In what follows all varieties will be assumed to be defined over an algebraically closed field of characteristic 0. Let X be a smooth projective variety of dimension $n\ge 3$ and let H be an ample line bundle on X. Let $K_X$ denote the canonical bundle of X and let $A^i(X)$ denote the codimension i Chow group of X. For a torsion-free sheaf E on X, the slope $\mu _H(E)$ is defined to be the following number

$$\begin{aligned} \mu _H(E):=\frac{c_1(E)\cdot H^{n-1}}{\mathrm{rk}\,E}. \end{aligned}$$

A torsion-free sheaf E on X is said to be $\mu$-stable(resp.$\mu$-semistable) with respect to H (or simply H-(semi)stable) if, for any coherent subsheaf $F\subset E$ with $0<\mathrm{rk}\,F<\mathrm{rk}\,E$, we have $\mu _H(F)<\mu _H(E)$ (resp.$\mu _H(F)\le \mu _H(E)$).

The discriminant $\Delta (E)\in A^2(X)$ of E is defined as follows.

$$\begin{aligned} \Delta (E)=2rc_2(E)-(r-1)c_1(E)^2. \end{aligned}$$

We set $\Delta _H(E):=\Delta (E)\cdot H^{n-2}$. We recall the following results concerning the restriction of $\mu$-(semi)stable sheaves to divisors ( [5]).

Proposition 1

Let X be a smooth projective variety X of dimension $n\ge 2$ and let H be a very ample line bundle on X. Let E be an H-semistable torsion-free sheaf of rank $r\ge 2$ on X. Let a be an integer with

$$\begin{aligned} \left( {\begin{array}{c}a+n\\ n\end{array}}\right) >\frac{1}{2}\left( \max \{\frac{r^2-1}{4},1\}H^n+1\right) \Delta _H(E)+1. \end{aligned}$$

Then, for general $D\in \vert aH\vert$, the restriction $E_{\vert D}$ is an $H_D$-semistable torsion-free sheaf.

We also need the following vanishing result due to H.Sun which has been proved by techniques of tilt stability ( [8, Corollary 1.9]).

Proposition 2

Let X be a smooth projective variety X of dimension $n\ge 2$ and H an ample line bundle on X. Let E be an H-semistable torsion-free sheaf of rank $r\ge 2$ and Chern classes $c_i(E)=c_i$ on X. Let

$$\begin{aligned} \overline{\Delta }_H(E):&=(c_1(E)\cdot H^{n-1})^2-2H^nr\mathrm{ch}_2(E)\cdot H^{n-2}\\&=(c_1(E)\cdot H^{n-1})^2+H^n(\Delta _H(E)-c_1(E)^2\cdot H^{n-2}). \end{aligned}$$

Then, for any integer l with

$$\begin{aligned} l>\frac{\overline{\Delta }_H(E)}{H^n}-\frac{\mu _H(E)}{H^n}, \end{aligned}$$

we have

$$\begin{aligned} H^{n-1}(X,E(K_X+lH))=0. \end{aligned}$$

Let X be a smooth projective variety of dimension $n\ge 2$. For a coherent sheaf E of rank r and Chern classes $c_i$ and a very ample line bundle H on X, we define the following numbers depending only on r, $c_i(E)$ ($i=1,\dots ,n$) and H.

$$\begin{aligned} a(E,H)&:=\min \{a\in {\mathbb {N}}\,\vert \left( {\begin{array}{c}a+n\\ n\end{array}}\right) >\frac{1}{2}\left( \max \{\frac{r^2-1}{4},1\}H^n+1\right) \Delta _H(E)+1\},\\ c(E,H)&:=\lfloor \frac{\overline{\Delta }_H(E)}{H^n}-\frac{\mu _H(E)}{H^n}\rfloor +1. \end{aligned}$$

Here, for a real number x, $\lfloor x\rfloor$ denotes the largest integer less than or equal to x. Let

$$\begin{aligned} a_0(E,H):=\lfloor \bigl \{\frac{n!}{2}(\max \{ \frac{r^2-1}{4},1\}H^n+1)\Delta _H(E)+1\bigr \}^{\frac{1}{n}}\rfloor +1. \end{aligned}$$

We see that $a_0:=a_0(E,H)\ge a(E,H)$ since

$$\begin{aligned} \left( {\begin{array}{c}a_0+n\\ n\end{array}}\right)>\frac{(a_0+1)^n}{n!}>\frac{a_0^n}{n!}>\frac{1}{2}(\max \{ \frac{r^2-1}{4},1\}H^n+1)\Delta _H(E)+1. \end{aligned}$$

Let X be a smooth projective threefold and let H be a very ample line bundle on X. Let $E_1$ be an H-semistable vector bundle of rank $r_1$ on X. In the rest of this section, we recall the upper bound of $\dim \mathrm{Ext}^1(E,E_1)$ obtained in [7] for H-semistable torsion-free sheaves E on X. We notice that we gave a bound of different type for sheaves on a Calabi-Yau threefold in [6].

For an H-semistable torsion-free sheaf E on X of rank $r\ge 2$ and Chern classes $c_i(E)=c_i$, let $\chi (E)$ denote the Euler characteristic of E. Then Riemann-Roch formula yields

$$\begin{aligned} \chi (E)&=\frac{1}{6}(c_1(E)^3-3c_1(E)\cdot c_2(E)+3c_3(E))+\frac{1}{12}c_1(E)\cdot (K_X^2+c_2(X))\\&+r\chi ({\mathcal {O}}_X). \end{aligned}$$

For a line bundle on L on X, we set $\alpha (E,L):=\chi (E\otimes L)-\chi (E)$. Then we have (cf. [7]):

$$\begin{aligned} \alpha (E,L)&=L\cdot \Bigl ( \frac{L^2}{6}+\frac{(2c_1(E)-rK_X)\cdot L}{4}+\frac{c_1(E)\cdot (c_1(E)-K_X)}{2}\\&-c_2(E)+\frac{r(K_X^2+c_2(X))}{12}\Bigr ). \end{aligned}$$

We set $E^{\prime }=E\otimes E_1^{\vee }$ and $l=\max \{a(E^{\prime },H),\,c(E^{\prime },H)\}$. We divide into the following six cases.

$$\begin{aligned} Case\, 1\text{- }1:\quad&(K_X+lH)\cdot H^2<0\quad \text {and}\quad \mu _H(E^{\prime })\ge -(K_X+lH)\cdot H^2 \\ Case\, 1\text{- }2:\quad&(K_X+lH)\cdot H^2<0\quad \text {and}\quad 0<\mu _H(E^{\prime })<-(K_X+lH)\cdot H^2 \\ Case\, 1\text{- }3:\quad&(K_X+lH)\cdot H^2<0\quad \text {and}\quad \mu _H(E^{\prime })\le 0 \\ Case\, 2\text{- }1:\quad&(K_X+lH)\cdot H^2\ge 0\quad \text {and}\quad \mu _H(E^{\prime })<-(K _X+lH)\cdot H^2 \\ Case\, 2\text{- }2:\quad&(K_X+lH)\cdot H^2\ge 0\quad \text {and}\quad -(K_X+lH)\cdot H^2\le \mu _H(E^{\prime })\le 0 \\ Case\, 2\text{- }3:\quad&(K_X+lH)\cdot H^2\ge 0\quad \text {and}\quad \mu _H(E^{\prime })>0. \\ \end{aligned}$$

Then we have the following bound for $\dim \mathrm{Ext}^1(E,E_1)$( [7, Theorem 3.3]).

Theorem 3

Let X, H, E and $E_1$ be as above and let $B_i:=B_i(E^{\prime },H,l)$. Then we have $\dim \mathrm{Ext}^1(E,E_1)\le B$ where

$$\begin{aligned} B={\left\{ \begin{array}{ll} B_1+B_2+B_3 &{} \text {in Case 2-2} \\ B_1+B_3 &{} \text {in Case 1-1 and Case 2-3} \\ B_2+B_3 &{} \text {in Case 1-3 and Case 2-1} \\ B_3&{} \text {in Case 1-2.} \\ \end{array}\right. } \end{aligned}$$

Here $B_j$ are defined as follows.

$$\begin{aligned} B_1(E,H,l)&=\displaystyle {rH^3\left( {\begin{array}{c}\frac{\mu _{H}(E)+(K_X+lH)\cdot H^2}{H^3}+f(r)+2\\ 2\end{array}}\right) }, \\ B_2(E,H,l)&=\displaystyle {rH^3\left( {\begin{array}{c}-\frac{\mu _{H}(E)}{H^3}+f(r)+2\\ 2\end{array}}\right) },\\ B_3(E,H,l)&=-\alpha (E(K_X),lH)\\&=-lH\cdot \Bigl ( \frac{l^2H^2}{6}+\frac{l(2c_1(E(K_X))-rK_X)\cdot H}{4}\\&\quad +\frac{c_1(E(K_X))\cdot (c_1(E(K_X))-K_X)}{2}-c_2(E(K_X))\\&\quad +\frac{r(K_X^2+c_2(X))}{12}\Bigr ). \end{aligned}$$

3 Effective bounds in dimension $n\ge 3$

We shall adopt the notations introduced in the previous section. The purpose of this section is to give effective bounds for several invariants of $\mu$-semistable sheaves on smooth projective variety. Let X be a smooth projective variety of dimension $n\ge 3$ and let H be a very ample line bundle on X. For integers $1\le i\le n-2$ and $l_1,\,l_2,\,\dots ,l_i$, we denote by $Y_i\in \vert l_1H\cap \dots \cap l_{i-1}H\vert$ a general smooth complete intersections of divisors $l_1H,\,l_2H,\,\dots ,l_iH$. Let $l_1(E,H):=\max \{a_0(E,H),c(E,H)\}$ and for $2\le i\le n-2$, define

$$\begin{aligned} l_i=l_i(E,H):=\max \{a_0(E_{\vert Y_{i-1}},H_{Y_{i-1}}),c(E_{\vert Y_{i-1}},H_{Y_{i-1}})\}. \end{aligned}$$

For $1\le j\le 3$, let $C_j=C_j(E,H):=B_j(E_{\vert Y},H_Y,l_{n-2})$ for general smooth threefold $Y=Y_{n-3}\in \vert l_1H\cap \dots \cap l_{n-3}H\vert$.

Proposition 4

Let $l_i$ and $C_j$ be as above. Then

• 1. For each $1\le i\le n-2$, $l_i$ depends only on r, $c_1(E)$, $c_2(E)$ and H.

• 2. For $1\le j\le 3$, $C_j$ depends only on r, $c_1(E)$, $c_2(E)$, $c_1(X)$, $c_2(X)$ and H.

Proof

For any integer $l>0$ and general smooth $Y\in \vert lH\vert$, let $\iota :Y\hookrightarrow X$ denote the inclusion. Then we have

$$\begin{aligned} \Delta _{H_Y}(E_{\vert Y})&=l\Delta _H(E),\\ \overline{\Delta }_{H_Y}(E_{\vert Y})&=l^2\overline{\Delta }_H(E). \end{aligned}$$

Hence we obtain

$$\begin{aligned} a_0(E_{\vert Y},H_Y)&=\lfloor \bigl \{\frac{(n-1)!l}{2}(\max \{ \frac{r^2-1}{4},1\}H^n+1)\Delta _H(E)+1\bigr \}^{\frac{1}{n-1}}\rfloor +1,\\ c(E_{\vert Y},H_Y)&=\lfloor \frac{l\overline{\Delta }_H(E)}{H^n}-\frac{\mu _H(E)}{H^n}\rfloor +1. \end{aligned}$$

By induction, the claim (1) follows immediately.

We notice that there exists the following exact sequence of tangent bundles on Y:

$$\begin{aligned} 0\rightarrow T_Y\rightarrow \iota ^*T_X\rightarrow N_{Y/X}\rightarrow 0 \end{aligned}$$

where $N_{Y/X}$ is the normal bundle of Y in X. Hence the total Chern class of $T_Y$ is given by

$$\begin{aligned} c(T_Y)=c(\iota ^*T_X)/c(N_{Y/X}) \end{aligned}$$

where

$$\begin{aligned} c(N_{Y/X})=\prod _{i=1}^{n-3}(1+l_iH_Y). \end{aligned}$$

Hence the claim (2) follows. $\square$

Let $E_1$ be an H-semistable vector bundle of rank $r_1$ on X. We are interested in estimating $\dim \mathrm{Ext}^1(E,E_1)$ from above for any H-semistable torsion-free sheaf E on X. Let $E^{\prime }=E\otimes E_1^{\vee }$. Then $E^{\prime }$ is H-semistable by [3, Theorem 3.1.4]. Let $l_i:=l_i(E^{\prime },H)$ for $1\le i\le n-2$, $l:=\sum _{i=1}^{n-2}l_i$ and $C_j:=C_j(E^{\prime },H)$ for $1\le j\le 3$. As in the case of threefolds, we consider the following six cases.

$$\begin{aligned} Case\, 1\text{- }1:\quad&(K_X+lH)\cdot H^{n-1}<0\quad \text {and}\quad \mu _H(E^{\prime })\ge -(K_X+lH)\cdot H^{n-1} \\ Case\, 1\text{- }2:\quad&(K_X+lH)\cdot H^{n-1}<0\quad \text {and} \quad 0< \mu _H(E^{\prime })<-(K_X+lH)\cdot H^{n-1} \\ Case\, 1\text{- }3:\quad&(K_X+lH)\cdot H^{n-1}<0\quad \text {and}\quad \mu _H(E^{\prime })\le 0 \\ Case\, 2\text{- }1:\quad&(K_X+lH)\cdot H^{n-1}\ge 0\quad \text {and}\quad \mu _H(E^{\prime })<-(K_X+lH)\cdot H^{n-1} \\ Case\, 2\text{- }2:\quad&(K_X+lH)\cdot H^{n-1}\ge 0\quad \text {and}\quad -(K_X+lH)\cdot H^{n-1}\le \mu _H(E^{\prime })\le 0 \\ Case\, 2\text{- }3:\quad&(K_X+lH)\cdot H^{n-1}\ge 0\quad \text {and}\quad \mu _H(E^{\prime })>0. \end{aligned}$$

Theorem 5

Let X, H, E and $E_1$, $l_i$ and $C_i$ be as above. Then we have $\dim \mathrm{Ext}^1(E,E_1)\le C$ where

$$\begin{aligned} C={\left\{ \begin{array}{ll} C_1+C_2+C_3 &{} \text {in Case 2-2} \\ C_1+C_3 &{} \text {in Case 1-1 and Case 2-3} \\ C_2+C_3 &{} \text {in Case 1-3 and Case 2-1} \\ C_3&{} \text {in Case 1-2} \\ \end{array}\right. } \end{aligned}$$

Proof

For any $1\le i\le n-3$ and general smooth $Y_{i+1}\in \vert l_{i+1}H_{Y_i}\vert$, we have the exact sequence on $Y_i$:

$$\begin{aligned} 0\rightarrow E_{\vert Y_i}(-l_{i+1}H_{Y_i})\rightarrow E_{\vert Y_i}\rightarrow E_{\vert Y_{i+1}}\rightarrow 0. \end{aligned}$$

By tensoring the above sequence with $E_1^{\vee }(K_{Y_i}+l_{i+1}H)$, we obtain the exact sequence

$$\begin{aligned} 0\rightarrow E^{\prime }(K_{Y_i})\rightarrow E^{\prime }(K_{Y_i}+l_{i+1}H)\rightarrow E^{\prime }(K_{Y_i}+l_{i+1}H)_{\vert Y_{i+1}}\rightarrow 0. \end{aligned}$$

By Proposition 1, $E^{\prime }_{\vert Y_i}$ is an $H_{Y_i}$-semistable sheaf on $Y_i$. Hence Proposition 2 yields $H^{n-i-1}(E^{\prime }(K_{Y_i} +l_{i+1}H))=0$. Then we obtain the surjection

$$\begin{aligned} H^{n-i-2}(E^{\prime }(K_X+l_{i+1}H)_{\vert Y_i})\rightarrow H^{n-i-1}(E^{\prime }(K_{Y_i})). \end{aligned}$$

Therefore we have $h^{n-i-1}(E^{\prime }(K_{Y_i}))\le h^{n-i-2}(E^{\prime }(K_{Y_i}+l_{i+1}H_{Y_i})_{\vert Y_{i+1}})$. Since Serre duality yields

$$\begin{aligned} H^{n-i-1}(E^{\prime }(K_{Y_i}))&\cong \mathrm{Ext}^1(E_{\vert Y_i},E_{1\vert Y_i})^{\vee },\\ H^{n-i-2}(E^{\prime }(K_{Y_i}+l_{i+1}H_{Y_i})_{\vert Y_{i+1}})&\cong \mathrm{Ext}^1(E_{\vert Y_{i+1}},E_{1\vert Y_{i+1}})^{\vee }, \end{aligned}$$

we obtain $\dim \mathrm{Ext}^1(E_{\vert Y_i},E_{1\vert Y_i})\le \dim \mathrm{Ext}^1(E_{\vert Y_{i+1}},E_{1\vert Y_{i+1}})$ for all $1\le i\le n-3$. It follows that $\dim \mathrm{Ext}^1(E,E_1)\le \dim \mathrm{Ext}^1(E_{\vert Y},E_{1\vert Y})$ for general smooth $Y=Y_{n-3}\in \vert l_1H\cap \dots \cap l_{n-3}H\vert$. We have

$$\begin{aligned} (K_Y+l_{n-2}H_Y)\cdot H_Y^2&=l^{\prime }(K_X+lH)\cdot H^{n-1},\\ \mu _{H_Y}(E^{\prime }_{\vert Y})&=l^{\prime }\mu _H(E^{\prime }) \end{aligned}$$

where $l=\sum _{i=1}^{n-2}l_i$ and $l^{\prime }=\prod _{i=1}^{n-3}l_i$. Hence, applying Theorem 3 to the threefold Y and the $H_Y$-semistable sheaf $E_{\vert Y}$, we obtain the claim for $\dim \mathrm{Ext}^1(E,E_1)$. $\square$

We notice that the constant C in the theorem above depends only on $r,\,c_i,\,c_i(X)$ and H and not on the choice of E, Y.

Corollary 6

Let X be a smooth projective variety of dimension $n\ge 3$ and let H be a very ample line bundle on X. Let $l_i=l_i(E(-K_X),H)$ and $m_i=l_i(E^{\vee },H)$ for $1\le i\le n-2$.

• 1. For any H-semistable torsion-free sheaf E on X of rank $r\ge 2$, $c_i(E)=c_i$, we have $h^{n-1}(E)\le \sum _{j=1}^3C_j$ where $C_j=C_j(E(-K_X),H)$.

• 2. For any H-semistable vector bundle E on X of rank $r\ge 2$, $c_i(E)=c_i$, we have $h^1(E)\le \sum _{j=1}^3D_j$ where $D_j=C_j(E^{\vee },H)$.

Proof

By Serre duality, we have $H^{n-1}(E)=\mathrm{Ext}^{n-1}({\mathcal {O}}_X,E)\cong \mathrm{Ext}^1(E,K_X)^{\vee }$. Hence, applying Theorem 5 to the sheaves E, $E_1=K_X$, we obtain (1). If E is a vector bundle, then $H^1(E)\cong \mathrm{Ext}^1(E^{\vee },{\mathcal {O}}_X)$. Hence we apply Theorem 3 to $E^{\vee }$ and $E_1={\mathcal {O}}_X$ and obtain (2). $\square$

Let X be a smooth projective variety of dimension $n\ge 2$ and let H be a very ample line bundle on X. For a coherent sheaf E on X, the Mukai vector v(E) of E is the element of the rational cohomology ring $H^*(X,{\mathbb {Q}}):=\oplus _{i=0}^4H^{2i}(X,{\mathbb {Q}})$ defined as follows.

$$\begin{aligned} v(E):=\mathrm{ch}(E)\cdot \sqrt{\mathrm{td}(X)} \end{aligned}$$

where $\mathrm{td}(X)$ denotes the Todd class of X. For given $v\in H^*(X,{\mathbb {Q}})$, let ${\mathcal {M}}(v)$ denote the moduli space of $\mu$-stable torsion-free sheaves with Mukai vector v. Let ${\mathcal {M}}(v)_0\subset {\mathcal {M}}(v)$ denote the open subscheme of $\mu$-stable locally free sheaves. Let $E_1$ be a $\mu$-stable rigid vector bundle on X. We define the Brill-Noether locus ${\mathcal {M}}(v)_{i,j}$ of type (i, j) as follows.

$$\begin{aligned} {\mathcal {M}}(v)_{i,j}:=\{E\in {\mathcal {M}}(v)\,\vert \,i=\dim \mathrm{Hom}(E_1,E)\, \text {and}\, j=\dim \mathrm{Ext}^1(E,E_1)\}. \end{aligned}$$

We are interested in the higher dimensional Brill-Noether problem concerning the existence of these loci. Theorem 5 yields the following

Corollary 7

Let X be a smooth projective variety of dimension $n\ge 3$ and let H be a very ample line bundle on X. Then ${\mathcal {M}}(v)_{i,j}$ is empty if $i\ge 0$ and $j>C$ where C is the constant in Theorem 5.

In general, we have the following inequality ( [3, Corollary 4.5.2])

$$\begin{aligned} \dim {\mathcal {M}}_{[E]}(v)\le \dim \mathrm{Ext}^1(E,E). \end{aligned}$$

We notice that effective bounds for $\dim {\mathcal {M}}_{[E]}(v)$ have been investigated for sheaves on a threefold in [7]. Applying Theorem 5 to $E_1=E$, we obtain the following result in dimension $n\ge 3$.

Proposition 8

Let X be a smooth projective variety of dimension $n\ge 3$ and let H be a very ample line bundle on X. For a $\mu$-stable vector bundle $E\in {\mathcal {M}}(v)_0$ on X, let $l_i:=l_i(\mathcal End E,H)$ $(1\le i\le n-2)$ and let $C_j:=C_j(\mathcal End E,H)$. Then we have $\dim {\mathcal {M}}_{[E]}(v)\le \sum _{j=1}^3C_j$.

4 Chern class inequalities on a fourfold

In this section we obtain an upper bound for the fourth Chern class $c_4(E)$ of $\mu$-semistable bundles on a smooth projective fourfold. First, we recall the following Riemann-Roch formula for sheaves on a fourfold.

Lemma 9

Let X be a smooth projective fourfold. Let E be a coherent sheaf of rank r with Chern classes $c_i$ on X. Then

$$\begin{aligned} \chi (E)&=\frac{1}{24}(c_1(E)^4-4c_1(E)^2\cdot c_2(E)+4c_1(E)\cdot c_3(E)+2c_2(E)^2-4c_4(E))\\&-\frac{1}{12}(c_1(E)^3-3c_1(E)\cdot c_2(E)+3c_3(E))\cdot K_X \\&+\frac{1}{24}(c_1(E)^2-2c_2(E))\cdot (K_X^2+c_2(X))-\frac{1}{24}c_1(E)\cdot K_X\cdot c_2(X)+r\chi ({\mathcal {O}}_X). \end{aligned}$$

We make explicit the constants $l_i(E,H)$ and $c_j(E,H)$ introduced in the previous section for sheaves E on a fourfold.

Lemma 10

Let X be a smooth projective fourfold and let H be a very ample line bundle on X. Let E be a torsion-free sheaf on X. Let $l_i:=l_i(E,H)$ for $i=1,\,2$ and $C_j:=C_j(E,H)$ for $1\le j\le 3$. Then

$$\begin{aligned} l_1&=\max \{\lfloor \bigl \{12(\max \{ \frac{r^2-1}{4},1\}H^4+1)\Delta _H(E)\bigr \}^{\frac{1}{4}}\rfloor , \lfloor \frac{\overline{\Delta }_H(E)}{H^4}-\frac{\mu _H(E)}{H^4}\rfloor \}+1, \\ l_2&=\max \{\lfloor \bigl \{3(\max \{ \frac{r^4-1}{4},1\}l_1H^4+1)l_1\Delta _H(E)\bigr \}^{\frac{1}{3}}\rfloor ,\lfloor \frac{l_1\overline{\Delta }_H(E)}{H^4}-\frac{\mu _H(E)}{H^4}\rfloor \}+1 \end{aligned}$$

and

$$\begin{aligned} C_1&=\displaystyle {rl_1H^4\left( {\begin{array}{c}\frac{\mu _{H}(E)+(K_X+(l_1+l_2)H)\cdot H^3}{H^4}+f(r)+2\\ 2\end{array}}\right) }, \\ C_2&=\displaystyle {rl_1H^4\left( {\begin{array}{c}-\frac{\mu _{H}(E)}{H^4}+f(r)+2\\ 2\end{array}}\right) },\\ C_3&=-l_1l_2H^2\cdot \Bigl ( \frac{l_2^2H^2}{6}+ \frac{l_2\{2c_1(E(K_X+l_1H))-r(K_X+l_1H)\}\cdot H}{4}\\&\quad +\frac{c_1(E(K_X+l_1H))\cdot \{c_1(E(K_X+l_1H))-(K_X+l_1H)\}}{2}\\&\quad -c_2(E(K_X+l_1H))+\frac{r\{(K_X+l_1H)^2+c_2(X)+l_1(K_X+l_1H)\cdot H\}}{12}\Bigr ). \end{aligned}$$

Proof

Let $\iota :Y\hookrightarrow X$ denote the inclusion map. Since we have $K_Y=(K_X+l_1H)_{\vert Y}$ and $c_2(Y)=\iota ^*(c_2(X)+l_1(K_X+l_1)H)\cdot H)$, the claim follows immediately. $\square$

Now we apply Theorem 5 to obtain a bound for the fourth Chern class of $\mu$-semistable bundles on a fourfold.

Theorem 11

Let X be a smooth projective fourfold and let H be a very ample line bundle on X. Let E be an H-semistable vector bundle E on X of rank $r\ge 2$, $c_i(E)=c_i$. Let $l_i=l_i(E(-K_X),H)$ and $m_i=l_i(E^{\vee },H)$ for $i=1,\,2$. We define

$$\begin{aligned} F&=\frac{1}{4}(c_1(E)^4-4c_1(E)^2\cdot c_2(E)+4c_1(E)\cdot c_3(E)+2c_2(E)^2)\\&\quad -\frac{1}{2}(c_1(E)^3-3c_1(E)\cdot c_2(E)+3c_3(E))\cdot K_X\\&\quad +\frac{1}{4}(c_1(E)^2-2c_2(E))\cdot (K_X^2+c_2(X))-\frac{1}{4}c_1(E)\cdot K_X\cdot c_2(X)+6r\chi ({\mathcal {O}}_X), \end{aligned}$$

$$\begin{aligned} C&=rl_1H^4\displaystyle {\left\{ \left( {\begin{array}{c}\frac{\mu _H(E(-K_X))+(K_X+(l_1+l_2)H)\cdot H^3}{H^4}+f(r)+2\\ 2\end{array}}\right) + \left( {\begin{array}{c}-\frac{\mu _H(E(-K_X))}{H^4}+f(r)+2\\ 2\end{array}}\right) \right\} }\\&\quad -l_1l_2H^2\cdot \Bigl ( \frac{l_2^2H^2}{6}+ \frac{l_2\{2c_1(E(l_1H))-r(K_X+l_1H)\}\cdot H}{4}\\&\quad +\frac{c_1(E(l_1H)\cdot \{c_1(E(l_1H)-(K_X+l_1H)\}}{2}\\&\quad -c_2(E(l_1H)+\frac{r\{(K_X+l_1H)^2+c_2(X)+l_1(K_X+l_1H)\cdot H\}}{12}\Bigr ) \end{aligned}$$

and

$$\begin{aligned} D&=rm_1H^4\displaystyle {\left\{ \left( {\begin{array}{c}\frac{\mu _H(E^{\vee })+(K_X+(m_1+m_2)H)\cdot H^3}{H^4}+f(r)+2\\ 2\end{array}}\right) + \left( {\begin{array}{c}-\frac{\mu _H(E^{\vee })}{H^4}+f(r)+2\\ 2\end{array}}\right) \right\} }\\&\quad -m_1m_2H^2\cdot \Bigl ( \frac{m_2^2H^2}{6}+ \frac{m_2\{2c_1(E^{\vee }(K_X+m_1H))-r(K_X+m_1H)\}\cdot H}{4}\\&\quad +\frac{c_1(E^{\vee }(K_X+m_1H))\cdot \{c_1(E^{\vee }(K_X+m_1H))-(K_X+m_1H)\}}{2}\\&\quad -c_2(E^{\vee }(K_X+m_1H))+\frac{r\{(K_X+m_1H)^2+c_2(X)+m_1(K_X+m_1H)\cdot H\}}{12}\Bigr ). \end{aligned}$$

Then we have $c_4(E)\le F+6(C+D)$.

Proof

By Corollary 6, we have $h^3(E)\le C:=\sum _{j=1}^3C_j$ and $h^1(E)\le D:=\sum _{j=1}^3D_j$ where $C_j=C_j(E(-K_X),H)$, $D_j=C_j(E^{\vee },H)$. Let $F=6\chi (E)+c_4(E)$. This yields $\chi (E)\ge -(C+D)$ and hence

$$\begin{aligned} c_4(E)&=F-6\chi (E)\\&\le F+6(C+D). \end{aligned}$$

Therefore the claim follows from Lemma 9 and Lemma 10. $\square$

We obtain the following bound in the case of abelian fourfolds.

Corollary 12

Let X be an abelian fourfold and let H be a very ample line bundle on X. Let E be an H-semistable vector bundle on X of rank $r\ge 2$ on X. Let $l_i=l_i(E,H)$ and $m_i=l_i(E^{\vee },H)$. Then we have $c_4(E)\le F+6(C+D)$ where

$$\begin{aligned} F&=\frac{1}{4}(c_1(E)^4-4c_1(E)^2c_2(E)+4c_1(E)c_3(E)+2c_2(E)^2),\\ C&=rl_1H^4\left\{ \displaystyle {\left( {\begin{array}{c}\frac{\mu _H(E)+(l_1+l_2)H^4}{H^4}+f(r)+2\\ 2\end{array}}\right) }+\displaystyle {\left( {\begin{array}{c}-\frac{\mu _H(E)}{H^4}+f(r)+2\\ 2\end{array}}\right) }\right\} \\&-l_1l_2H^2\cdot \Bigl ( \frac{l_2^2H^2}{6}+ \frac{l_2\{2c_1(E(l_1H))-rl_1H\}\cdot H}{4}\\&+\frac{c_1(E(l_1H))\cdot \{c_1(E(l_1H))-l_1^2H\}}{2} -c_2(E(l_1H))+\frac{rl_1^2H^2}{6}\Bigr ), \\ D&=rm_1H^4\left\{ \displaystyle {\left( {\begin{array}{c}\frac{-\mu _H(E)+(m_1+m_2)H)\cdot H^3}{H^4}+f(r)+2\\ 2\end{array}}\right) }+ \displaystyle {\left( {\begin{array}{c}\frac{\mu _H(E)}{H^4}+f(r)+2\\ 2\end{array}}\right) }\right\} \\&-m_1m_2H^2\cdot \Bigl ( \frac{m_2^2H^2}{6}- \frac{m_2\{2c_1(E^{\vee }(m_1H))-rm_1H\}\cdot H}{4}\\&+\frac{c_1(E^{\vee }(m_1H))\cdot \{c_1(E^{\vee }(m_1H))-m_1^2H\}}{2}-c_2(E^{\vee }(m_1H))+\frac{rm_1^2H^2\}}{6}\Bigr ). \end{aligned}$$

We notice that there cannot exist an analogous upper bound for $c_4(E)$ for H-semistable torsion-free sheaves E on a smooth projective fourfold X. Indeed, the following result holds in arbitrary dimension.

Proposition 13

Let X be a smooth projective variety of dimension $n\ge 2$ and H an ample line bundle on X. Assume that n is even (resp. odd). Then there does not exist an upper (resp. lower) bound for $c_n(E)$ for H-semistable torsion-free sheaves E on X in terms of r, $c_i$ for $1\le i\le n-1$, H and $c_i(X)$.

Proof

Let E be an H-semistable torsion-free sheaf on X of rank r, $c_i(E)=c_i$ and any point $p\in X$, let $E_p$ denote the kernel of the natural evaluation map $E\rightarrow {\mathcal {O}}_p$. Then $E_p$ is an H-semistable torsion-free sheaf of rank r, $c_i(E_p)=c_i$ for $1\le i\le n-1$ and $c_n(E_p)=c_n(E)-(-1)^{n+1}(n-1)!$. Therefore the claim follows by choosing arbitrarily many points of X. $\square$