Surface of general type

In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this class.
Classification.
Gieseker showed that there is a coarse moduli scheme for surfaces of general type; this means that for any fixed values of the Chern numbers formula_1 there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers. It remains a very difficult problem to describe these schemes explicitly, and there are few pairs of Chern numbers for which this has been done (except when the scheme is empty). There are some indications that these schemes are in general too complicated to write down explicitly: the known upper bounds for the number of components are very large, some components can be non-reduced everywhere, components may have many different dimensions, and the few pieces that have been studied explicitly tend to look rather complicated.
The study of which pairs of Chern numbers can occur for a surface of general type is known as "" and there is an almost complete answer to this question. There are several conditions that the Chern numbers of a minimal complex surface of general type must satisfy:
Many (and possibly all) pairs of integers satisfying these conditions are the Chern numbers for some complex surface of general type.
By contrast, for almost complex surfaces, the only constraint is: 
and this can always be realized.
Examples.
This is only a small selection of the rather large number of examples of surfaces of general type that have been found. Many of the surfaces of general type that have been investigated lie on (or near) the edges of the region of possible Chern numbers. In particular Horikawa surfaces lie on or near the "Noether line", many of the surfaces listed below lie on the line formula_7 the minimum possible value for general type, and surfaces on the line formula_8 are all quotients of the unit ball in C2 (and are particularly hard to find).
Surfaces with χ=1.
These surface which are located in the "lower left" boundary in the diagram have been studied in detail. For these surfaces with second Chern class can be any integer from 3 to 11. Surfaces with all these values are known; a few of the many examples that have been studied are:
Canonical models.
 proved that the multicanonical map φ"nK" for a complex surface of general type is a birational isomorphism onto its image whenever "n"≥5, and showed that the same result still holds in positive characteristic. There are some surfaces for which it is not a birational isomorphism when "n" is 4.
These results follow from Reider's theorem.