Monads and regularity of vector bundles on projective varieties (Q2469314)

In their previous paper [Nagoya Math. J. 186, 119--155 (2007; Zbl 1134.14010)], the authors extended, using helix theory, the Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties \(X\) of dimension \(n\) admitting an exceptional collection \((E^0_1,\dots ,E^0_{{\alpha}_0}, \dots ,E^n_1,\dots ,E^n_{{\alpha}_n})\) consisting of (exceptional) vector bundles which generate the derived category \({\mathcal D}:= {\text{D}}^b(\text{Coh}X)\) and which can be packed together into \(n+1\) blocks \({\mathcal E}_i=(E^i_1,\dots ,E^i_{{\alpha}_i})\) having the property that \({\text{Hom}}_{\mathcal D}(E^i_j,E^i_k)=0\) if \(j\neq k\), \(i=0,\dots ,n\). Examples of such varieties are the projective space \({\mathbb P}^n\) with the collection \(({\mathcal O}_{\mathbb P},{\mathcal O}_{\mathbb P}(1),\dots , {\mathcal O}_{\mathbb P}(n))\) (in which case the regularity defined by Costa and Miró-Roig coincides with the Castelnuovo-Mumford regularity), the smooth hyperquadric \(Q_n\subset {\mathbb P}^{n+1}\), with the collection \(({\mathcal O}_Q,{\mathcal O}_Q(1),\dots ,{\mathcal O}_Q(n-1),{\mathcal E}_n)\), where \({\mathcal E}_n=({\Sigma}_1(n-1),{\Sigma}_2(n-1))\) for \(n\) even and \({\mathcal E}_n=\Sigma (n-1)\) for n odd (\(\Sigma \), \({\Sigma}_1\), \({\Sigma}_2\) being the spinor bundles), the Grassmannians, the multiprojective spaces etc. In the paper under review, the authors bound the regularity of a vector bundle \(E\) on \({\mathbb P}^n\) or on \(Q_n\) defined as the cohomology of a monad \[ 0\rightarrow \bigoplus {\mathcal O}(a_i)\rightarrow \bigoplus {\mathcal O}(b_j)\rightarrow \bigoplus {\mathcal O}(c_l) \rightarrow 0 \] in terms of the integers \(a_i\), \(b_j\), \(c_l\). They also improve this bound in the particular case where \(E\) is a mathematical instanton bundle on \(Q_n\), \(n\) odd.