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On the non-existence of orthogonal instanton bundles on \(\mathbb P^{2n+1}\). (English) Zbl 1200.14085
Instanton bundles are vector bundles \(E\) of rank \(2n\), on a projective space of odd dimension \(2n+1\), arising as the middle cohomology of a monad. There are examples of instanton bundles which are symplectic, in the sense that there exists an isomorphism \(\alpha: E\to E^*\) such that \(\alpha = -\alpha^*\). In particular, every instanton bundle on \(\mathbb P^3\) is symplectic. One could rise the question about the existence of orthogonal instanton bundles, i.e. bundles endowed with an isomorphism \(\alpha\) as above, with \(\alpha =\alpha^*\). The authors close the question, by showing that no instanton bundles are orthogonal.

MSC:
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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