even when they are equal they may be greater than the irregularity (the dimension of the Picard variety). Invariants of surfaces Hodge numbers and Kodaira dimension The most important invariants of a compact complex surfaces used in the classification can be given in terms of the dimensions of various coherent sheaf cohomology groups. The basic ones are the plurigenera and the Hodge numbers defined as follows: K is the canonical line bundle whose sections are the holomorphic 2-forms. are called the plurigenera. They are birational invariants, i.e., invariant under blowing up. Using Seiberg–Witten theory, Robert Friedman and John Morgan showed that for complex manifolds they only depend on the underlying oriented smooth 4-manifold. For non-Kähler surfaces the plurigenera are determined by the fundamental group, but for Kähler surfaces there are examples of surfaces that are homeomorphic but have different plurigenera and Kodaira dimensions. The individual plurigenera are not often used; the most important thing about them is their growth rate, measured by the Kodaira dimension. is the Kodaira dimension: it is (sometimes written −1) if the plurigenera are all 0, and is otherwise the smallest number (0, 1, or 2 for surfaces) such that is bounded. Enriques did not use this definition: instead he used the values of and . These determine the Kodaira dimension given the following correspondence: where is the sheaf of holomorphic i-forms, are the Hodge numbers, often arranged in the Hodge diamond: By Serre duality and The Hodge numbers of a complex surface depend only on the oriented real cohomology ring of the surface, and are invariant under birational transformations except for which increases by 1 under blowing up a single point. If the surface is Kähler then and there are only three independent Hodge numbers. If the surface is compact then equals or Invariants related to Hodge numbers There are many invariants that (at least for complex surfaces) can be written as linear combinations of the Hodge numbers, as follows: Betti numbers: defined by In characteristic p > 0 the Betti numbers are defined using l-adic cohomology and need not satisfy these relations. Euler characteristic or Euler number: The irregularity is defined as the dimension of the Picard variety and the Albanese variety and denoted by q. For complex surfaces (but not always for surfaces of prime characteristic) The geometric genus: The arithmetic genus: The holomorphic Euler characteristic of the trivial bundle (usually differs from