1
$\begingroup$

Let $S$ be a $K3$ surface and $H$ an ample line bundle on it. Fix a Mukai vector $v\in H^*(S,\mathbb Z)$. If $v$ is primitive, it turns out that Gieseker stability w.r.t.$H$ coincides with Gieseker semistability w.r.t. $H$. Does the same hold for $\mu_H$-stability, too?

$\endgroup$

2 Answers 2

2
$\begingroup$

If the rank and degree (wrt $H$) are coprime, then $\mu$-stability and $\mu$-semistability coincide. The argument is the same as for Gieseker-stability, but simpler.

$\endgroup$
1
$\begingroup$

The answer is no, as $\mu_H$-stability only sees the rank and the first Chern class: if there are two Mukai vectors $v_1, v_2$ with $rk(v_1) + rk(v_2) = rk(v)$ and $H . c_1(v_1) + H.c_1(v_2) = H.c_1(v)$, then a short exact sequence $0 \to E_1 \to E \to E_2 \to 0$ where each $E, E_i$ has the corresponding Mukai vector $v, v_i$ will make $E$ strictly semistable.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.