Exotic characteristic classes of quaternionic bundles
Let \((E^{2n},\omega)\) denote a complex symplectic vector space. The author associates to any Hermitian metric \(\gamma\) on \(E^{2n}\) a positive compatible quaternionic structure \(J:=a(-a^ 2)^{-1/2},\) where \(a: E^{2n}\to E^{2n}\) is the unique antilinear isomorphism with \(\gamma (ax,y)=-\overline{\omega (x,y)}.\) Let \(\pi\) : \(E\to V\) be a complex vector bundle with two quaternionic structures \(J_ 1\) and \(J_ 2\), and quaternionic metric connections \(\nabla^ 1\), \(\nabla^ 2\) with local equations \(\nabla^ ac_ i=\gamma_ i^{aj}c_ j.\) The author defines the cohomology classes \[ \xi_{2s+1}(J^ 1,J^ 2):=[{\mathcal D}_{2s+1}(\nabla^ 1,\nabla^ 2)]\in H^{4s+1}(V,{\mathbb{R}}), \] where \[ {\mathcal D}_ k(\nabla^ 1,\nabla^ 2):=(i/2\pi)^ k/(k- 1)!\int^{1}_{0}\delta^{j_ 1...j_ k}_{i_ 1...i_ k}(\gamma_{j_ 1}^{2i_ 1}-\gamma_{j_ 1}^{1i_ 1})\wedge \Pi^{i_ 2}_{tj_ 2}\wedge...\wedge \Pi^{i_ k}_{tj_ k}dt, \] \[ \Pi^ i_{tj}:=(1-t)\Pi^ i_{0j}+t\Pi^ j_{i1}+t(1- t)(\gamma_ j^{2h}-\gamma_ j^{1h})\wedge (\gamma_ h^{2i}- \gamma_ h^{1i}), \] and \(\Pi_ 0\), \(\Pi_ 1\) are the curvature forms of \(\nabla^ 1\), \(\nabla^ 2\), respectively. The author in this way defines to any pair of complex symplectic structures a sequence of exotic characteristic classes \(\xi_{2s+1}(\omega_ 1,\omega_ 2)\in H^{4s+1}(M,{\mathbb{R}}),\) which vanish if the structures are differentiably homotopically related via complex symplectic structures \(\omega_ t\), and have the properties \(\xi_{2s+1}(\omega_ 1,\omega_ 2)=- \xi_{2s+1}(\omega_ 2,\omega_ 1),\quad \xi_{2s+1}(\omega_ 1,\omega_ 2)+\xi_{2s+1}(\omega_ 2,\omega_ 3)+\xi_{2s+1}(\omega_ 3,\omega_ 1)=0.\) If \(E=T_{hol}T^*_{hol}M,\) and M is Kählerian, then E has a canonical symplectic structure. If \(\beta\) is a \(C^{\infty}\)-automorphism of \(T_{hol}M\), then a second complex symplectic structure can be obtained. This leads to exotic characteristic classes which retract to M and can be represented by certain differential forms.
complex vector bundle with two quaternionic structures
scientific article; zbMATH DE number 4173924