Automorphisms of Projective Varieties

This lecture delves into the study of automorphisms of projective varieties, focusing on the isomorphism between complements and the extension to automorphisms of PN. The instructor discusses observations regarding the dimensions of X and Y, providing examples to illustrate the concepts. The lecture explores the history of the conjecture for n equal to 2 and presents counterexamples in higher dimensions. Key observations and the main theorem by the instructor and Pierre-Marie are highlighted, addressing the existence of examples that do not satisfy certain conditions. The lecture concludes with insights into piecewise isomorphism and the implications of the results in different characteristic fields.