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?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
If the rank and degree (wrt $H$) are coprime, then $\mu$stability and $\mu$semistability coincide. The argument is the same as for Giesekerstability, but simpler. 


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. 

