Irregularity of a surface

In mathematics, the irregularity of a complex surface X is the Hodge number h^0,1= dim H¹(O_X), usually denoted by q (Wolf P. Barth, Klaus Hulek & Chris A.M. Peters et al. 2004). 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 (Bombieri & Mumford 1977, p.26), which is the same in characteristic 0 but can be smaller in positive characteristic.

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 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 in general dimension the Hodge number h^0,1 = dim H¹(O_X) is called irregularity q.

For non-singular complex projective (or Kähler) surfaces the following numbers are all equal:

For surfaces in positive characteristic, or for non-Kähler complex surfaces, the numbers above need not all be equal.

Poincaré (1910) proved that for complex projective surfaces the dimension of the Picard variety is equal to the Hodge number h^0,1, and the same is true for all compact Kähler surfaces. The irregularity of smooth compact Kähler surfaces is invariant under bimeromorphic transformations.

For general compact complex surfaces the two Hodge numbers h^1,0 and h^0,1 need not be equal, but h^0,1 is either h^1,0 or h^1,0+1, and is equal to h^1,0 for compact Kähler surfaces.

Over fields of positive characteristic, the relation between q (defined as the dimension of the Picard or Albanese variety), and the Hodge numbers h^0,1 and h^1,0 is more complicated, and any two of them can be different.

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