Irregularity of a surface

In mathematics, the irregularity of a complex surface X is the Hodge number ${\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})}$, usually denoted by q.^[1] The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety, which is the same in characteristic 0 but can be smaller in positive characteristic.^[2]

The name "irregularity" comes from the fact that for the first surfaces investigated in detail, the smooth complex surfaces in P³, the irregularity happens to vanish. The irregularity then appeared as a new "correction" term measuring the difference ${\displaystyle p_{g}-p_{a}}$ of the geometric genus and the arithmetic genus of more complicated surfaces. Surfaces are sometimes called regular or irregular depending on whether or not the irregularity vanishes.

For a complex analytic manifold X of general dimension, the Hodge number ${\displaystyle h^{0,1}=\dim H^{1}({\mathcal {O}}_{X})}$ is called the irregularity of ${\displaystyle X}$, and is denoted by q.