In this article we provide a classification of the projective transformations in P SL(n + 1, C) considered as automorphisms of the complex projective space P n C . Our classification is an interplay between algebra and dynamics. Just as in the case of isometries of CAT (0)-spaces, this is given by means of three types of transformations, namely: elliptic, parabolic and loxodromic. We describe the dynamics in each case, more precisely we determine the corresponding Kulkarni's limit set, the equicontinuity region, minimal sets, the discontinuity region and maximal regions where the corresponding cyclic group acts properly discontinuously. We also provide, in each case, some equivalent ways to classify the projective transformations.

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In this article we show that Bers' simultaneous uniformization as well as the Köebe's retrosection theorem are not longer true for discrete groups of projective transformations acting on the complex projective space.

We complete the classification of type preserving representations of Deligne-Mostow lattices with 3-fold symmetry into PGL(3, C) started in [FPUP21a]. In particular, we show local rigidity for all the representations where the generators we chose are of the same type as the generators of the Deligne-Mostow lattices. We use formal computations in SAGE and MAPLE to obtain the results. The code files are available on GitHub ([FPUP21b]).

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