must satisfy: (the Bogomolov–Miyaoka–Yau inequality) (the Noether inequality) Most pairs of integers satisfying these conditions are the Chern numbers for some complex surface of general type. Examples: The simplest examples are the product of two curves of genus at least 2, and a hypersurface of degree at least 5 in P3. There are a large number of other constructions known. However, there is no known construction that can produce "typical" surfaces of general type for large Chern numbers; in fact it is not even known if there is any reasonable concept of a "typical" surface of general type. There are many other examples that have been found, including most Hilbert modular surfaces, fake projective planes, Barlow surfaces, and so on. See also List of algebraic surfaces References – the standard reference book for compact complex surfaces ; ( softcover) – including a more elementary introduction to the classification External links le superficie algebriche is an interactive visualisation of the Enriques--Kodaira classification, by Pieter Belmans and Johan Commelin Complex surfaces Birational geometry Algebraic surfaces Mathematical classification systems