- If \(\mathcal{E}\) is a locally free sheaf of rank \(r\) on a (nonsingular) curve \(C\), then there is a sequence \[ 0=\mathcal{E}_0 \subseteq \mathcal{E}_1 \subseteq \ldots \subseteq \mathcal{E}_r=\mathcal{E} \] of subsheaves such that \(\mathcal{E}_i / \mathcal{E}_{i-1}\) is an invertible sheaf for each \(i=1, \ldots, r\). We say that \(\mathcal{E}\) is a successive extension of invertible sheaves.
Hint: Use (II, Ex. 8.2).
- Show that this is false for varieties of dimension \(\geqslant 2\). In particular, the sheaf of differentials \(\Omega\) on \(\PP^2\) is not an extension of invertible sheaves.
Solution
1. Let \(\mathcal E\) be locally free of rank \(r\geq2\) on the nonsingularprojective curve \(C\). There is an invertible subsheaf\[ \mathcal L\hookrightarrow\mathcal E \]whose quotient is locally free of rank \(r-1\).
Fix a very ample invertible sheaf \(\OO_C(1)\). For \(n\gg0\), Serre’s theorem makes \[ \mathcal E(n)=\mathcal E\tensor\OO_C(n) \] generated by global sections. Since \[ \operatorname{rank}\mathcal E(n)=r>1=\dim C, \] Exercise II.8.2, proved on P-AGH282GENERICSECTION, gives a nowhere-vanishing global section and an exact sequence \[ 0\longrightarrow\OO_C \longrightarrow\mathcal E(n) \longrightarrow\mathcal F \longrightarrow0, \] where \(\mathcal F\) is locally free of rank \(r-1\). Tensoring by \(\OO_C(-n)\) gives \[ 0\longrightarrow\OO_C(-n) \longrightarrow\mathcal E \longrightarrow\mathcal F(-n) \longrightarrow0. \] Thus one may take \[ \mathcal L=\OO_C(-n). \]
2. Every locally free sheaf of rank \(r\) on \(C\) is a successive extensionof invertible sheaves.
Proceed by induction on \(r\). For \(r=1\) there is nothing to prove.
Assume \(r\geq2\). By step 1 there is an exact sequence \[ 0\longrightarrow\mathcal L \longrightarrow\mathcal E \xrightarrow{\pi}\mathcal Q \longrightarrow0 \] with \(\mathcal L\) invertible and \(\mathcal Q\) locally free of rank \(r-1\). By induction, choose a filtration \[ 0=\mathcal Q_0\subseteq\mathcal Q_1\subseteq\cdots \subseteq\mathcal Q_{r-1}=\mathcal Q \] whose successive quotients are invertible.
Define \[ \mathcal E_0=0, \qquad \mathcal E_1=\mathcal L, \qquad \mathcal E_{i+1}=\pi^{-1}(\mathcal Q_i) \quad(1\leq i\leq r-1). \] Then \[ \mathcal E_1/\mathcal E_0\cong\mathcal L \] and, for \(1\leq i\leq r-1\), \[ \mathcal E_{i+1}/\mathcal E_i \cong \mathcal Q_i/\mathcal Q_{i-1}. \] Every quotient on the right is invertible. Hence \[ 0=\mathcal E_0\subseteq\mathcal E_1\subseteq\cdots \subseteq\mathcal E_r=\mathcal E \] is the required filtration. This proves part (a).
3. The cotangent bundle on \(\PP^2\) has total Chern class\[ c\qty(\Omega^1_{\PP^2})=1-3H+3H^2, \]where\[ H=c_1\qty(\OO_{\PP^2}(1)). \]
The cotangent Euler sequence is \[ 0\longrightarrow\Omega^1_{\PP^2} \longrightarrow\OO_{\PP^2}(-1)^{\oplus3} \longrightarrow\OO_{\PP^2} \longrightarrow0. \] Multiplicativity of total Chern classes in a short exact sequence gives \[ c\qty(\Omega^1_{\PP^2}) = c\qty(\OO(-1)^{\oplus3}) =(1-H)^3. \] Since \(H^3=0\) in the Chow ring of \(\PP^2\), \[ (1-H)^3=1-3H+3H^2. \]
4. The bundle \(\Omega^1_{\PP^2}\) is not an extension of invertiblesheaves.
Because \(\Omega^1_{\PP^2}\) has rank two, a successive-extension filtration would give an exact sequence \[ 0\longrightarrow\mathcal L \longrightarrow\Omega^1_{\PP^2} \longrightarrow\mathcal M \longrightarrow0 \] with \(\mathcal L,\mathcal M\) invertible. The Picard group of \(\PP^2\) is generated by \(\OO(1)\) [@Har10a, Proposition II.6.4 and Corollary II.6.16], so for some integers \(a,b\), \[ \mathcal L\cong\OO(a), \qquad \mathcal M\cong\OO(b). \] Multiplicativity of Chern classes and step 3 would then give \[ 1-3H+3H^2 = (1+aH)(1+bH) = 1+(a+b)H+abH^2. \] Thus \[ a+b=-3, \qquad ab=3. \] The integers \(a,b\) would therefore be roots of \[ t^2+3t+3=0, \] whose discriminant is \[ 9-12=-3. \] This is impossible over \(\ZZ\). Hence \[ \boxed{\Omega^1_{\PP^2}\text{ is not a successive extension of line bundles}.} \]
5. For every integer \(n\geq2\), there is a smooth projective \(n\)-fold witha locally free sheaf that is not a successive extension of invertible sheaves.
4. For \(n>2\), put\[ Y=\PP^2\times\PP^{n-2} \]and let\[ \mathcal G=\operatorname{pr}_1^*\Omega^1_{\PP^2}. \]Suppose \(\mathcal G\) admitted a filtration with invertible successivequotients. Restrict it to a closed fibre\[ \PP^2\times\{y\}\cong\PP^2. \]Because every successive quotient is locally free, restriction preserves theshort exact sequences in the filtration. It would therefore give a filtrationof\[ \mathcal G|_{\PP^2\times\{y\}} \cong \Omega^1_{\PP^2} \]with invertible successive quotients, contradicting step
4. Thus thecurve statement does not extend to any dimension \(n\geq2\), proving part (b).
6. Q.E.D.
Steps <1>1–<1>2 prove part (a), and steps <1>3–<1>5 prove part (b), including the stated example on \(\PP^2\).