Irregularity of a surface
Encyclopedia
In mathematics, the irregularity of a complex surface X is the Hodge number h0,1= dim H1(OX), usually denoted by q . 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.
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 surfaces in positive characteristic, or for non-Kaehler complex surfaces, the numbers above need not all be equal.
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 Kaehler surfaces. The irregularity of smooth compact Kaehler 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 Kaehler surfaces.
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).
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. . In positive characteristic neither Hodge number is always bounded by the other:
showed that it is possible for h1,0 to vanish while
h0,1 is positive, while showed that for Enriques surface
s in characteristic 2 it is possible for h0,1 to vanish while h1,0 is positive.
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 charactersitic 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. 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.
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
Geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties and complex manifolds.-Definition:...
and the arithmetic genus
Arithmetic genus
In mathematics, the arithmetic genus of an algebraic variety is one of some possible generalizations of the genus of an algebraic curve or Riemann surface.The arithmetic genus of a projective complex manifold...
of more complicated surfaces. Surfaces are sometimes called regular or irregular depending on whether or not the irregularity vanishes.
Complex surfaces
For non-singular complex projective (or Kaehler) surfaces the following numbers are all equal:- The irregularity
- The dimension of the Albanese varietyAlbanese varietyIn mathematics, the Albanese variety A, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. In other words there is a morphism from the variety V to its Albanese variety A, such that any morphism from V to an...
- 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 genusGeometric genusIn algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties and complex manifolds.-Definition:...
and the arithmetic genusArithmetic genusIn mathematics, the arithmetic genus of an algebraic variety is one of some possible generalizations of the genus of an algebraic curve or Riemann surface.The arithmetic genus of a projective complex manifold...
.
For surfaces in positive characteristic, or for non-Kaehler complex surfaces, the numbers above need not all be equal.
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 Kaehler surfaces. The irregularity of smooth compact Kaehler 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 Kaehler 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 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).
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. . In positive characteristic neither Hodge number is always bounded by the other:
showed that it is possible for h1,0 to vanish while
h0,1 is positive, while showed that for Enriques surface
Enriques surface
In mathematics, Enriques surfaces, discovered by , are complex algebraic surfacessuch that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square...
s in characteristic 2 it is possible for h0,1 to vanish while h1,0 is positive.
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
Pierre Cartier (mathematician)
Pierre Cartier is a mathematician. An associate of the Bourbaki group and at one time a colleague of Alexander Grothendieck, his interests have ranged over algebraic geometry, representation theory, mathematical physics, and category theory....
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 charactersitic 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. 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.