Irregularity of a surface
In mathematics, the irregularity of a complex surface X is the Hodge number h0,1= dim H1(OX), 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 pg − pa 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 h0,1 = dim H1(OX) is called irregularity q.
For non-singular complex projective (or Kähler) surfaces the following numbers are all equal:
- The irregularity
- The dimension of the Albanese variety
- The dimension of the Picard variety
- The Hodge number h0,1 = dim(H1(Ω0))
- The Hodge number h1,0 = dim(H0(Ω1))
- The difference pg − pa of the geometric genus and the arithmetic genus.
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 h0,1, 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 h1,0 and h0,1 need not be equal, but h0,1 is either h1,0 or h1,0+1, and is equal to h1,0 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 h0,1 and h1,0 is more complicated, and any two of them can be different.
There is a canonical map from a surface F to its Albanese variety A which induces a homomorphism from the cotangent space of the Albanese variety (of dimension q) to H1,0(F). Igusa (1955) showed that this is injective, so that q ≤ h1,0, but shortly after found a surface in characteristic 2 with h1,0= h0,1 = 2 and Picard variety of dimension 1, so that q can be strictly less than both Hodge numbers. (Igusa 1955b). In positive characteristic neither Hodge number is always bounded by the other: Serre (1956) showed that it is possible for h1,0 to vanish while h0,1 is positive, while Mumford (1961) showed that for Enriques surfaces in characteristic 2 it is possible for h0,1 to vanish while h1,0 is positive.
Grothendieck (1961, p.23-24) gave a complete description of the relation of q to h0,1 in all characteristics. The dimension of the tangent space to the Picard scheme (at any point) is equal to h0,1. In characteristic 0 a result of Cartier shows that all groups schemes of finite type are non-singular, so the dimension of their tangent space is their dimension. On the other hand, in positive characteristic it is possible for a group scheme to be non-reduced at every point so that the dimension is less than the dimension of any tangent space, which is what happens in Igusa's example. Mumford (1966, lecture 27) shows that the tangent space to the Picard variety is the subspace of H0,1 annihilated by all Bockstein operations from H0,1 to H0,2. So the irregularity q is equal to h0,1 if and only if all these Bockstein operations vanish.
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