aCM bundles on a general \(K3\) surface of degree 2 (Q6597506)

The paper studies aCM bundles on a general \(K3\) surface. An aCM (arithmetically Cohen-Macaulay) bundle on a smooth projective manifold \(X\) is a vector bundle \(E\) such that \(H^i(X,E(m))=0\) for all \(0<i<\dim X\) and \(m\in\mathbb{Z}\). The paper focuses on the case when \(X\) is a smooth \(K3\) surface with \(\text{Pic}(X)\cong\mathbb{Z}H\) such that \(H\) is an ample divisor and \(H^2=2\). Then \(E\) is aCM iff \(h^1(E(mH))=h^1(E^{\vee}(mH))=0\) for all \(m\in\mathbb{Z}_{\geq 0}\).\N\NBy the previous results of the author and his collaborators, for a stable bundle \(E\) with Mukai vector \(v:=(r,dH,a)\) and any positive integer \(p\), if \(H^1(X,E)=0\) and \(\frac{d}{r}>\frac{p}{2p^2+1}\), then \(H^1(X,E(pH))=0\). Therefore under some suitable assumption on \(r,d\) and \(a\), the key step is to prove \(H^1(X,E)=0\).\N\NThe main strategy to show \(H^1(E)=0\) is to use the Fourier-Mukai transform. Let \(E_0\) be a stable vector bundle of Mukai vector \(v_0:=(2,H,1)\). Then it is known that \(H^1(E_0)=0\). Let \(\mathcal{E}=\text{Ker}(E_0^{\vee}\boxtimes E_0\rightarrow \mathcal{O}_{\triangle})\in\text{Coh}(X\times X)\). Let \(\Phi:=\Phi_{X\rightarrow X}^{\mathcal{E}}\) be the Fourier-Mukai transform with kernel \(\mathcal{E}\). Let \(v':=\Phi(v)\). Under suitable assumption on \(r,d\) and \(a\), for a general stable bundle \(E\) of class \(v\), \(F:=\Phi(E)\) is a stable bundle of class \(v'\) and \(F\) is of rank \(\leq 3\). We have the following exact sequence\N\[\N0\rightarrow E_0^{\oplus \langle v,v_0\rangle}\rightarrow E\rightarrow F\rightarrow 0,\N\]\Ninduced by the exact triangle\N\[\N\mathcal{O}_{\triangle}[-2]\rightarrow E_0\boxtimes E_0^{\vee}\rightarrow \mathcal{E}^{\vee}\rightarrow \mathcal{O}_{\triangle}[-1].\N\]\NSince the rank of \(F\) is small and it has been shown before by the author and his collaborators that \(H^1(F)=0\), we get that \(H^1(E)=0\).\N\NThe main result of the paper states that for \(r\geq 2,0\leq d\leq r\), there is a stable aCM sheaf iff\N\N(i) \(d^2+(r-d)^2\leq 2d^2-2ra\leq 2r^2+\min\{ad^2,2(r-d)^2\}\) and\N\N(ii) \(v\neq (2,0,-1),(2,0,-1)e^H\).\N\NThe sufficiency is because the argument above works under this condition. Conversely assume there is a stable aCM sheaf \(E\) of Mukai vector \(v\). Then \(E\) has to be a bundle and (ii) holds since otherwise there is no stable bundle of Mukai vector \(v\). (i) can be shown by using the double cover \(\pi:X\rightarrow \mathbb{P}^2\). \(\pi_*(E)\) is also an aCM sheaf on \(\mathbb{P}^2\) and moreover it is a direct sum of line bundles by splitting criterion of Horrocks. Therefore by an direct computation one gets (i).\N\NIn the end, the paper classifies aCM bundles of rank 2 and 3 on \(X\) and gives some brief discussion on the case \(H^2=n\geq 4\).