Monads on projective varieties (Q1750830)

Let $X$ be a projective variety and let $0\to A\to B\to C\to 0$ be a complex of coherent sheaves which is exact on the left and right. This is called a monad and we say that the cohomology in the middle is represented by this monad. \textit{G. Fløystad} [Commun. Algebra 28, No. 12, 5503--5516 (2000; Zbl 0977.14007)] studied these in projective spaces $\mathbb{P}^n$, where $A=\mathcal{O}_{\mathbb{P}^n}(-1)^a$, $B=\mathcal{O}_{\mathbb{P}^n}^b$ and $C=\mathcal{O}_{\mathbb{P}^n}(1)^c$ and proved various existence and non-existence theorems. The paper under review generalizes these to more general varieties. Let me reproduce one of the results to give a flavor. Let $X$ be a smooth projective variety of dimension $n$ with an embedding in some projective space and assume we have a monad as above, where $A,B,C$ are restrictions from projective spaces of the above bundles. If the cohomology of this monad is of rank less than $n$, then it must be of rank $n-1$ with $n$ odd and $a=c$. Conversely, if $X$ is an odd dimensional non-singular variety embedded in a projective space and it is arithmetically Cohen-Macaulay, the for any $a=c\geq 1$, one has such a monad with vector bundle cohomology.