Spectral covers and a specific exact short sequence

Question

I have a question about the spectral cover construction of Friedman, Morgan, and Witten (typically used to map a description in heterotic string theory into F-theory). I realise this is a highly specialised topic; I am asking in the hope that there might be some specialists around. I also realise this may be more appropriate in the mathematics StackExchange, and may post a question there later.

Specifically I have a question about the more mathematical FMW paper, `Vector bundles over elliptic fibrations': https://arxiv.org/abs/alg-geom/9709029

Consider the second short exact sequence on page 72, of sheaves on the elliptic fibration $Z$ (the one involving $V_{A,a}$ sheaves). The three sheaves appearing in the first two slots of the sequence are I think supposed to all be vector bundles (the $V_{A,a}$ sheaves should be vector bundles since they are supposed to correspond to the spectral cover description). But the sheaf in the third slot is, I think, zero (that is, not a trivial bundle, but actually just zero) away from the subspace $D$, and hence is not a vector bundle. How then could both the left and the middle sheaves in the sequence be vector bundles? Additionally, away from the subspace $D$, do we then have an isomorphism between the sheaves in the first two slots? And in the subspace $D$ (where the final sheaf is not just zero), do we just have the ordinary quotient relation between the three sheaves?

Edit:

The short exact sequence in question is, $$0\to V_{A,a}(n)\to V_{A,a}(n-1)\oplus \pi^*L^a\to(\pi^*L^a)|D\to0\,.$$ The space where these bundles (or sheaves) live is an elliptic fibration $\pi:Z\to B$ (and $L^a$, where it appears, is being pulled back by $\pi^*$ to the space $Z$). The $|D$ denotes restriction to a subspace $D$ of $Z$. Here $V_{A,a}(m)$ is a sheaf, and specifically is I believe supposed to be a vector bundle for both values of $m$ seen in the sequence. The final term in the sequence is I think not a vector bundle (it is 0 away from the subspace $D$), and this gives rise to my questions above. (Essentially this is a question about exact sequences of sheaves, but as it appeared in a physics context I thought I'd try my luck finding someone on here who is familiar with this spectral cover construction.)

Answer 1

The quotient of locally free sheaves is not necessarily locally free, i.e. the first two terms being locally free does not force the third to be locally free, cf. this math.SE post.

You have no guarantee that, given a map $f: E\to F$ of vector bundles, that the quotient $\mathrm{coker}(E\to F)$ (which we would also like to write as $F/\mathrm{im}(E)$) exists as a vector bundle. In other words, the category of vector bundles is not an Abelian category, it does not have all kernels and cokernels. To see this, simply choose any map of vector bundles whose rank on the fibers jumps somewhere. This shows that you should not expect to be able to lift an exact sequence of sheaves to a sequence of vector bundles unless you know all sheaves to be locally free, since the category of sheaves is Abelian so maps of two locally free sheaves will always have kernels and cokernels as sheaves that are not kernels and cokernels as locally free sheaves.