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.
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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
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.
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
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
Then, for any integer l with
we have
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.
Here, for a real number x, \(\lfloor x\rfloor \) denotes the largest integer less than or equal to x. Let
We see that \(a_0:=a_0(E,H)\ge a(E,H)\) since
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
For a line bundle on L on X, we set \(\alpha (E,L):=\chi (E\otimes L)-\chi (E)\). Then we have (cf. [7]):
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.
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
Here \(B_j\) are defined as follows.
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
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
Hence we obtain
By induction, the claim (1) follows immediately.
We notice that there exists the following exact sequence of tangent bundles on Y:
where \(N_{Y/X}\) is the normal bundle of Y in X. Hence the total Chern class of \(T_Y\) is given by
where
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.
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
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 \(Y_i\):
By tensoring the above sequence with \(E_1^{\vee }(K_{Y_i}+l_{i+1}H)\), we obtain the exact sequence
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
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
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
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.
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.
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])
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
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
and
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
and
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
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
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. \(\square \)
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Acknowledgements
The author was supported by JSPS KAKENHI Grant Number JP20K03544.
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Communicated by Indranil Biswas.
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Nakashima, T. Cohomology bounds and Chern class inequalities for stable sheaves on a smooth projective variety. Indian J Pure Appl Math 54, 789–796 (2023). https://doi.org/10.1007/s13226-022-00297-8
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DOI: https://doi.org/10.1007/s13226-022-00297-8