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.

Complex surfaces

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 h^0,1 = dim(H¹(Ω⁰))
• The Hodge number h^1,0 = dim(H⁰(Ω¹))
• The difference p_g − p_a 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 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.

Positive characteristic

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.

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 H^1,0(F). Igusa (1955) showed that this is injective, so that q ≤ h^1,0, but shortly after found a surface in characteristic 2 with h^1,0= h^0,1 = 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 h^1,0 to vanish while h^0,1 is positive, while Mumford (1961) showed that for Enriques surfaces in characteristic 2 it is possible for h^0,1 to vanish while h^1,0 is positive.

Grothendieck (1961, p.23-24) gave a complete description of the relation of q to h^0,1 in all characteristics. The dimension of the tangent space to the Picard scheme (at any point) is equal to h^0,1. 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 H^0,1 annihilated by all Bockstein operations from H^0,1 to H^0,2. So the irregularity q is equal to h^0,1 if and only if all these Bockstein operations vanish.

References

• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
• Bombieri, Enrico; Mumford, David (1977), "Enriques' classification of surfaces in char. p. II", Complex analysis and algebraic geometry, Tokyo: Iwanami Shoten, pp. 23–42, MR 0491719
• Grothendieck, Alexander (1961), Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki 221
• Igusa, Jun-ichi (1955), "A fundamental inequality in the theory of Picard varieties", Proceedings of the National Academy of Sciences of the United States of America, 41 (5): 317–320, doi:10.1073/pnas.41.5.317, ISSN 0027-8424, JSTOR 89124, MR 0071113
• Igusa, Jun-ichi (1955b), "On some problems in abstract algebraic geometry", Proc. Natl. Acad. Sci. U.S.A., National Academy of Sciences, 41 (11): 964–967, doi:10.1073/pnas.41.11.964, JSTOR 89189, MR 0074085
• Mumford, David (1961), "Pathologies of modular algebraic surfaces", American Journal of Mathematics, The Johns Hopkins University Press, 83 (2): 339–342, doi:10.2307/2372959, ISSN 0002-9327, JSTOR 2372959, MR 0124328
• Mumford, David (1966), Lectures on curves on an algebraic surface, Annals of Mathematics Studies, 59, Princeton University Press, ISBN 978-0-691-07993-6, MR 0209285
• Poincaré, Henri (1910), "Sur les courbes tracées sur les surfaces algébriques", Annales Scientifiques de l'École Normale Supérieure, 3, 27: 55–108
• Serre, Jean-Pierre (1958), "Sur la topologie des variétés algébriques en caractéristique p", Symposium internacional de topología algebraica, Universidad Nacional Autónoma de México and UNESCO, Mexico City, pp. 24–53, MR 0098097