Question

Hi, i'm stuck on the following, please can someone help? Let $E$ be a complex holomorphic vector bundle of rank r over a compact kahler manifold $M$, let me indicate $\mathcal{E}$ the associated locally free sheaf of $E$, let $\mathcal{F}$ be a coherent subsheaf of $\mathcal{E}$ of rank $0< p < r$ such that $\frac{\mathcal{E}}{\mathcal{F}}$ is torsion free. The inclusion map $j:\mathcal{F}\rightarrow \mathcal{E}$ induces a homomorphism of sheaves $det(j): det(\mathcal{F})=({\Lambda}^p\mathcal{F})^{** } \rightarrow ({\Lambda}^p\mathcal{E})^{**}$ (with * i mean the dual sheaf), $det(j)$ is injective outside the singularity set $S_{n-1}(\mathcal{F})\subset M$ (the set of points in which $\mathcal{F}$ is not free), writing the sequence: $0 \rightarrow ker(det(j))\rightarrow det(\mathcal{F})\rightarrow (\Lambda^{p}\mathcal{E})^{ ** }$ why the sheaf $ker(det(j))$ is a torsion sheaf?

Thank you in advance

Answer 1

Let's assume for simplicity that $M$ is a smooth, complex, projective variety.

The set of points where the coherent subsheaf $\mathcal{F}$ is not locally free is a proper closed subset of $M$ (Hartshorne, Algebraic Geometry, Chapter II, ex. 5.8), so the stalk of $ker(det(j))$ at the generic point is zero, i.e. it is a torsion sheaf.

Moreover, you can say more. Indeed, since $\mathcal{E}$ is locally free and $\mathcal{E} /\mathcal{F}$ is torsion-free, it follows that $\mathcal{F}$ is a reflexive sheaf (Hartshorne, Stable Reflexive Sheaves, Theorem 1.1), so it is locally free except along a closed subset of codimension $\geq 3$ (same reference, Corollary 1.4).

In particular, if $M$ is a curve or a surface then $ker(det(j))$ is zero.