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Monads on projective varieties. (English) Zbl 1412.14012
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. 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.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14J10 Families, moduli, classification: algebraic theory 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli