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?
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2 Answers
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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.
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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.
answered Mar 4, 2011 at 9:24