a lattice. Kodaira surfaces These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles. The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces. Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order k = 1, 2, 3, 4, 6, then the plurigenera Pn are 1 if n is divisible by k and 0 otherwise. Hodge diamond: Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the fibers by Z acting as multiplication by powers of some complex number z. This gives a primary Kodaira surface. Enriques surfaces These are the complex surfaces such that q = 0 and the canonical line bundle is non-trivial, but has trivial square. Enriques surfaces are all algebraic (and therefore Kähler). They are quotients of K3 surfaces by a group of order 2 and their theory is similar to that of algebraic K3 surfaces. Invariants: The plurigenera Pn are 1 if n is even and 0 if n is odd. The fundamental group has order 2. The second cohomology group H2(X, Z) is isomorphic to the sum of the unique even unimodular lattice II1,9 of dimension 10 and signature −8 and a group of order 2. Hodge diamond: Marked Enriques surfaces form a connected 10-dimensional family, which has been described explicitly. In characteristic 2 there are some extra families of Enriques surfaces called singular and supersingular Enriques surfaces; see the article on Enriques surfaces for details. Hyperelliptic (or bielliptic) surfaces Over the complex numbers these are quotients of a product of two elliptic curves by a finite group of automorphisms. The finite group can be Z/2Z, Z/2Z + Z/2Z, Z/3Z, Z/3Z + Z/3Z, Z/4Z, Z/4Z + Z/2Z, or Z/6Z, giving seven families of such surfaces. Over fields of characteristics 2 or 3 there are some extra families given by taking quotients by a non-etale group scheme; see the article on hyperelliptic surfaces for details. Hodge diamond: Surfaces of Kodaira dimension 1 An elliptic surface is a surface equipped with an elliptic fibration (a surjective