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 Killing structure Jacobi operator of a real hypersurface in a complex projective space J. Korean Math. Soc.Published online September 23, 2020 Juan de Dios Pérez Universidad de Granada Abstract : We prove non-existence of real hypersurfaces with Killing structure Jacobi operator in complex projective spaces. We also classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Killing with respect to the $k$-th generalized Tanaka-Webster connection. Keywords : g-Tanaka-Webster connection; complex projective space; real hypersurface; Killing structure Jacobi operator MSC numbers : 53C15, 53B25 Full-Text :