the Euler number e defined above): By Noether's formula it is also equal to the Todd genus The signature of the second cohomology group for complex surfaces is denoted by : are the dimensions of the maximal positive and negative definite subspaces of so: c2 = e and are the Chern numbers, defined as the integrals of various polynomials in the Chern classes over the manifold. Other invariants There are further invariants of compact complex surfaces that are not used so much in the classification. These include algebraic invariants such as the Picard group Pic(X) of divisors modulo linear equivalence, its quotient the Néron–Severi group NS(X) with rank the Picard number ρ, topological invariants such as the fundamental group π1 and the integral homology and cohomology groups, and invariants of the underlying smooth 4-manifold such as the Seiberg–Witten invariants and Donaldson invariants. Minimal models and blowing up Any surface is birational to a non-singular surface, so for most purposes it is enough to classify the non-singular surfaces. Given any point on a surface, we can form a new surface by blowing up this point, which means roughly that we replace it by a copy of the projective line. For the purpose of this article, a non-singular surface X is called minimal if it cannot be obtained from another non-singular surface by blowing up a point. By Castelnuovo's contraction theorem, this is equivalent to saying that X has no (−1)-curves (smooth rational curves with self-intersection number −1). (In the more modern terminology of the minimal model program, a smooth projective surface X would be called minimal if its canonical line bundle KX is nef. A smooth projective surface has a minimal model in that stronger sense if and only if its Kodaira dimension is nonnegative.) Every surface X is birational to a minimal non-singular surface, and this minimal non-singular surface is unique if X has Kodaira dimension at least 0 or is not algebraic. Algebraic surfaces of Kodaira dimension may be birational to more than one minimal non-singular surface, but it is easy to describe the relation between these minimal surfaces. For example, P1 × P1 blown up at a point is isomorphic to P2 blown up twice. So to classify all compact complex surfaces up to birational isomorphism it is (more or less) enough to classify the minimal non-singular ones. Surfaces of Kodaira dimension −∞ Algebraic surfaces of Kodaira dimension