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The paper begins by overviewing the basic facts on geometric exceptionalcollections. Then, we derive, for any coherent sheaf $\cF$ on a smoothprojective variety with a geometric collection, two spectral sequences: thefirst one abuts to $\cF$ and the second one to its cohomology. The main goal ofthe paper is to generalize Castelnuovo-Mumford regularity for coherent sheaveson projective spaces to coherent sheaves on smooth projective varieties $X$with a geometric collection $\sigma $. We define the notion of regularity of acoherent sheaf $\cF$ on $X$ with respect to $\sigma$. We show that the basicformal properties of the Castelnuovo-Mumford regularity of coherent sheavesover projective spaces continue to hold in this new setting and we show that incase of coherent sheaves on $\PP^n$ and for a suitable geometric collection ofcoherent sheaves on $\PP^n$ both notions of regularity coincide. Finally, wecarefully study the regularity of coherent sheaves on a smooth quadrichypersurface $Q_n \subset \PP^{n+1}$ ($n$ odd) with respect to a suitablegeometric collection and we compare it with the Castelnuovo-Mumford regularityof their extension by zero in $\PP^{n+1}$.

Geometric collections and Castelnuovo–Mumford regularity