# Example of reflexive coherent sheaves that are not locally free

The question is in the title, do you have simple examples of reflexive sheaves on a complex manifold $$X$$ (it can be Kähler if you prefer, or even smooth algebraic) that are not locally free (i.e. vector bundles) ?

I know that such a sheaf $$\mathcal{F}$$ is such that $$\mathrm{S}(\mathcal{F}) = \{x \in X|\mathcal{F}_x$$ is not free$$\}$$ has codimension greater or equal than $$3$$ because $$\mathcal{F}$$ is a $$2$$-syzygy sheaf hence we must look for $$\dim(X) \geqslant 3$$. I also know that if $$\mathcal{F}$$ has rank $$1$$, $$\det(\mathcal{F}) = \mathcal{F}^{**}$$ is a line bundle so $$\mathcal{F}$$ reflexive $$\Leftrightarrow \mathcal{F}$$ locally free in this case hence we must look for $$\mathrm{rk}(\mathcal{F}) \geqslant 2$$.

It is easy to find torsion-free sheaves that are not locally free, by taking sheaves of ideals generating non-empty codimension $$\geqslant 2$$ analytic/algebraic subsets but it won't make a refexive sheaf because it has rank $$1$$. Do you have a simple example, like on $$\mathbb{C}^3$$ or $$\mathbb{P}^3(\mathbb{C})$$ for example ? And do you have example of reflexive stable coherent sheaves that are not locally free in the Kähler case ?

Take $$R$$ be the co-ordinate ring of algebraic functions on $$\mathbb{C}^3$$, so $$R$$ is just the polynomial ring in three variables $$x,y,z$$. Let $$M= Re_1\oplus Re_2\oplus Re_3/xe_1+ye_2+ze_3$$. Then, $$M$$ is reflexive, but not locally free.