Algebraic and geometric dynamics in several complex variables.

Abstract

We discuss various algebraic, geometric and dynamical properties of the holomorphic self-maps of complex projective manifolds. <italic>Self-maps of</italic> <blkbd>P2</blkbd> <italic>with invariant elliptic curves</italic>. If a regular self-map of <blkbd>P2</blkbd> leaves invariant an elliptic plane curve <italic>C</italic>, the closure of the backward orbit of any point on <italic>C</italic> equals the Julia set. For smooth cubics, two types of self-maps are discussed: tangent processes, and elementary maps. For singular elliptic curves, invariants are defined at the singular points, and calculated for several families of curves: duals of smooth cubics, and elliptic quartics with two singular points. <italic>Self-maps of ruled surfaces</italic>. We give a systematic description of the self-maps of ruled surfaces. The discussion is based on the rigidity of the curves with negative or zero self-intersection. The configurations of completely invariant curves is listed, and an application to the dynamics of elementary maps of <blkbd>P2</blkbd> is deduced. <italic>Completely invariant hypersurfaces in projective spaces</italic>. We discuss the structure of those hypersurfaces that are completely invariant for some rational self-map of <blkbd>Pn</blkbd> . This involves the study of essential hypersurfaces, Bottcher divisors, degenerate essential components and stars. <italic>Self-maps of projective bundles</italic>. Given a self-map of a projective bundle over <blkbd>Pn</blkbd> , we study its geometry (fiber-degree, algebraic degree, lifting to the dual vector bundle) and its dynamics (Green function, Fatou components, Julia set).