Construction of \(c_ 2\)-self-dual bundles on a quaternionic projective space
Higher-dimensional analogues of (anti)-self-dual connections were given independently by \textit{T. Nitta} [Tôhoku Math. J., II. Ser. 40, 425-440 (1988; Zbl 0663.53055)] and \textit{M. M. Capria} and \textit{S. M. Salamon} [ Nonlinearity 1, 517-530 (1988; Zbl 0681.53037)] on a quaternionic Kähler manifold (\(c_1\), \(c_2\), and \(c_3\) are self dual connections). In this paper a necessary condition for the existence of a bundle \(F\) on \(\mathbb{H} P^n\) which admits a \(c_2\)-self-dual \(Sp(r)\) connection (\(r\geq 2\)) with Chern class \(c(F)=(1-x)^{-k}\), where \(x\) is the standard generator of \(H^4(\mathbb{H} P^n,{\mathbb{Z}})\), is given. This condition is expressed in terms of the vanishing of the cohomology group \(H^1({\mathbb{P}}^{2n+1},\tilde{F}(p))=0\) for \(p\leq -2\), where \(\tilde{F}\) is the pullback bundle of \(F\) to \({\mathbb{P}}^{2n+1}\). The construction of \(F\) is based on Bellinson's spectral sequence argument. In the paper algebraic geometrical techniques are used.
scientific article; zbMATH DE number 902845