Nonorientable manifolds, complex structures, and holomorphic vector bundles (Q5960120)

The authors are interested to generalize a lot of notions from orientable to nonorientable manifolds. At first they define a generalization of the notion of an almost complex structure on a nonorientable smooth manifold \(M\) of even dimension. Let \(M\) be a connected nonorientable \(C^\infty \)-manifold of dimension \(2d\). Let \(\xi\) denote the orientation line bundle over \(M\) and let \(L=\wedge^{2d} T^*M\) denote the top exterior power of the cotangent bundle. If \(L_0=M \times\mathbb{R}\), then there exists a canonical isomorphism \(\rho: \xi^{\otimes 2} \to L_0\). A nonorientable almost complex structure \(J\) on \(M\) is a smoothly varying bundle isomorphism \(J:TM\to TM\otimes \xi\) such that the composition \[ TM @>J>> TM\otimes \xi @>J\otimes Id_\xi>> TM\otimes \xi^{ \otimes 2} @>Id_{TM}\otimes\rho>> TM \] coincides with the isomorphism of \(TM\) defined by multiplication with \(-1\). It is proved that if \(M\) is a real manifold of dimension \(2d\) equipped with a nonorientable almost complex structure, then \(d\) must be odd. A notion of integrability of this almost complex structure is defined and the Kähler form condition for a \(\xi\)-valued 2-form \(\omega\) (as a section of the complex vector bundle \(\xi\otimes_\mathbb{R} \otimes \wedge^2T^*M\otimes_\mathbb{R} \mathbb{C})\) is extended. Then the notion of complex vector bundle is generalized to the nonorientable context, as a nonorientable complex vector bundle. This is a real vector bundle of even rank such that the almost complex structure of a fiber is given up to the sign. Such bundles have generalized Chern classes. These are elements of the cohomology group \(H^{2*}(M, \xi^{\otimes *})\) of the tensor power of the local system defined by the orientation bundle. The notion of a holomorphic vector bundle over a manifold \(M\) equipped with an integrable nonorientable almost complex structure is defined. Stable vector bundles and Einstein-Hermitian connections are also generalized. It is shown that a generalized holomorphic vector bundle on a compact nonorientable Kähler manifold admits an Einstein-Hermitian connection if and only if it is polystable.