A numerical invariant of a non-singular projective algebraic variety $X$, equal to the dimension of its Picard variety. If the ground field has characteristic zero (or, more general, if the Picard scheme of $X$ is reduced), then the irregularity coincides with the dimension of the first cohomology space $H^1(X,\mathcal O_X)$ with coefficients in the structure sheaf.
A variety with non-zero irregularity is called irregular, and a variety with zero irregularity — regular. Sometimes the $i$-th irregularity of a complete linear system $|D|$ on a variety $X$ is defined as
$$\sigma^i(D)=\dim H^i(X,\mathcal O_X(D)),$$
where $1\leq i\leq\dim X$.
|[a1]||R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001|
Irregularity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irregularity&oldid=32658