A local holomorphic version of the fundamental theorem of projective geometry

The classical result in affine and projective geometry on maps taking straight lines to straight lines is generalized to locally defined holomorphic maps taking complex lines in a certain family to complex lines. This is used to investigate CR-maps of Levi-degenerate real CR-hypersurfaces in a complex space. Namely, the CR-maps between tube hypersurfaces in $\mathbb C^3$ whose Levi form has eigenvalue zero are fully characterized.

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