Vector bundles over curves

Question

On page 5 of "Mixed twistor structures", C. Simpson writes

Recall that a strict subsheaf of a vector bundle over a curve is determined by its restriction to any Zariski open set

How is this true? Firstly, every vector bundle $E$ is a strict subsheaf of another vector bundle, just take $E \oplus \mathcal O_C$, for $C$ the curve. Therefore I'm confused by his insistence on strictness.

Now if we take a trivialising open cover $\{U_i\}$ of the curve $C$ for the vector bundle $E$ and restrict $E$ to $U_i$, how could this determine $E$ uniquely?

Once my confusion has been cleared up, I would like to know if there are related results for higher-dimensional (projective) varieties.

Answer 1

Most probably what Simpson means is that a strict subsheaf of a fixed vector bundle is determined by its restriction to any nonempty Zariski open set. Indeed, if $E$ is the fixed vector bundle, $j \colon U \to C$ is the embedding of a nonempty Zariski open set, and $F \subset j^*E$ is a strict subsheaf, extend the embedding to an exact sequence $$ 0 \to F \to j^*E \to G \to 0 $$ and define $$ \tilde{F} := \operatorname{Ker}(E \to j_*G), $$ where the morphism $E \to j_*G$ is the adjoint of $j^*E \to G$. Then $\tilde{F} \subset E$ is the strict subsheaf in $E$ determined by $F \subset j^*E$.

Answer 2

In general if you have an open immersion $j:U\subset X$ and a closed immersion $i:Z\to X$ sich that $X=Z\cup U$,then giving a sheaf $F$ over $X$ is the same as giving a triple $F_1,F_2,\phi$ where $F_1$ is a sheaf over $U$,$F_2$ is a sheaf over $Z$ and $\phi$ is a morphism $\phi:F_2\to i^*j_* F_1$. I learn this from page 73 of Milne book on etale cohomology but this should be pretty standard.

Now back to the problem, in the case you want, you have a fixed vector bundle $E$ and a Zariski open $U$, because $C$ is a curve $C-U$ is union of some points so $i^* E$ is a trivial bundle.so $F_2$ should be a trivial bundle with a rank smaller than $E$. this show that in general $F\subset E$ can not determined by restriction of $F$ to the $U$(which we called $F_1$) because you have different choices for $F_2$. but if you want $F$ to be a sub-vectorbundle then the rank of $F_2$ should be equal to the rank of $F_1$ so $F_1$ determines $F_2$ and $\phi_F$ is determanined by $\phi_E$. In summary the true statement is this:

a Sub-vector bundle of a vector bundle $E$ over a curve $C$ is determined by its restriction to any open Zariski subset.