In mathematics, the irregularity of a complex surface X is the Hodge number
[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.
The irregularity then appeared as a new "correction" term measuring the difference
For a complex analytic manifold X of general dimension, the Hodge number
Henri Poincaré 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.
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).
and Picard variety of dimension 1, so that q can be strictly less than both Hodge numbers.
[5][6] Alexander Grothendieck gave a complete description of the relation of q to
The dimension of the tangent space to the Picard scheme (at any point) is equal to
[7] In characteristic 0 a result of Pierre Cartier showed that all groups schemes of finite type are non-singular, so the dimension of their tangent space is their dimension.
Mumford 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.