Limit of stable vector bundles.

Question

Let $S$ be a $K3$ surface and $H$ be an ample line bundle on it. Given a flat family of coherent sheaves on $S$ whose generic point is a $\mu_H$-stable vector bundle, what can I say about the non-generic points? Are they always $\mu_H$-semistable and torsion free? I think that this is equivalent to requiring the existence of a compactification of the moduli space of $\mu_H$-stable vector bundles with fixed Chern classes, whose boundary points are $\mu_H$-semistable torsion free sheaves.

Answer 1

I think the answer to your question is no, because of the following example. For the details, see [Huybrechts-Lehn, The geometry of moduli space of sheaves, Section 5.3].

Let $\pi \colon X \to \mathbb{P}^1$ be an elliptic $K3$ surface with irreducible fibres and assume that there is a section $\sigma \subset X$. By adjunction $\sigma$ is a $(-2)$-curve. Let $f$ be the class of a fibre of $\pi$ and set $H= H_m:= \sigma + (m+3) f$.

Then the Nakai-Moishezon criterion implies that $H$ is ample for $ m \geq 0$. Moreover, for $m >>0$, the $\mu_{H}$-semistability of a rank $2$ vector bundle $E$ such that $c_1(E) \cdot f \equiv 1$ (mod $2$) is equivalent to its $\mu_H$-stability.

Now let us consider an extension of the form

$$0 \to \mathcal{O}_X (f) \to E \to \mathcal{O}_X( \sigma - 2f) \to 0.$$

One proves that this extension is $\mu_H$-stable if and only if it does not split. Moreover, $c_1(E) \cdot f = (\sigma -f) \cdot f=1$, so $\mu_H$-stability and $\mu_H$-semistability are equivalent in this case (at least if $m$ is large enough).

Finally, $\textrm{Ext}^1(\mathcal{O}_X( \sigma - 2f), \mathcal{O}_X (f) ) \cong \mathbb{C}^2$, and the zero element of this cohomology group corresponds to the trivial extension. So if one takes a disk $\mathcal{D} \subset \mathbb{C}^2$ centered at the origin, there is a flat family

$$\{E_t\}_{t \in \mathcal{D}}$$

of sheaves on $X$ such that $E_t$ is $\mu_H$-stable for $t \neq 0$, whereas $E_0=\mathcal{O}_X (f) \oplus \mathcal{O}_X( \sigma - 2f)$ is $\mu_H$-instable. In fact, the inclusion $ \mathcal{O}_X( \sigma - 2f) \hookrightarrow E_0$ provides a destabilizing sub-bundle of $E_0$.