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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
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