are 0. Hodge diamond: Surfaces of Kodaira dimension 0 These surfaces are classified by starting with Noether's formula For Kodaira dimension 0, K has zero intersection number with itself, so Using we arrive at: Moreover since κ = 0 we have: combining this with the previous equation gives: In general 2h0,1 ≥ b1, so three terms on the left are non-negative integers and there are only a few solutions to this equation. For algebraic surfaces 2h0,1 − b1 is an even integer between 0 and 2pg. For compact complex surfaces 2h0,1 − b1 = 0 or 1. For Kähler surfaces 2h0,1 − b1 = 0 and h1,0 = h0,1. Most solutions to these conditions correspond to classes of surfaces, as in the following table: K3 surfaces These are the minimal compact complex surfaces of Kodaira dimension 0 with q = 0 and trivial canonical line bundle. They are all Kähler manifolds. All K3 surfaces are diffeomorphic, and their diffeomorphism class is an important example of a smooth spin simply connected 4-manifold. Invariants: The second cohomology group H2(X, Z) is isomorphic to the unique even unimodular lattice II3,19 of dimension 22 and signature −16. Hodge diamond: Examples: Degree 4 hypersurfaces in P3(C) Kummer surfaces. These are obtained by quotienting out an abelian surface by the automorphism a → −a, then blowing up the 16 singular points. A marked K3 surface is a K3 surface together with an isomorphism from II3,19 to H2(X, Z). The moduli space of marked K3 surfaces is connected non-Hausdorff smooth analytic space of dimension 20. The algebraic K3 surfaces form a countable collection of 19-dimensional subvarieties of it. Abelian surfaces and 2-dimensional complex tori The two-dimensional complex tori include the abelian surfaces. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic. The algebraic ones are exactly the 2-dimensional abelian varieties. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Criteria to be a product of two elliptic curves (up to isogeny) were a popular study in the nineteenth century. Invariants: The plurigenera are all 1. The surface is diffeomorphic to S1 × S1 × S1 × S1 so the fundamental group is Z4. Hodge diamond: Examples: A product of two elliptic curves. The Jacobian of a genus 2 curve. Any quotient of C2 by