On the non-existence of orthogonal instanton bundles on \(\mathbb P^{2n+1}\). (English) Zbl 1200.14085

Farnik, Łucja; Frapporti, Davide; Marchesi, Simone

Matematiche 64, No. 2, 81-90 (2009).

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.

Reviewer: Luca Chiantini (Siena)

MSC:

14J60

Vector bundles on surfaces and higher-dimensional varieties, and their moduli

Keywords:

instanton bundles