Geometric genus as dimension of a variety

Question

Given a smooth complex algebraic variety $X$ of dimension $n$, Hodge number $h^{1,0}(X)$ (irregularity) is the dimension of Albanese variety $A(X)$ of $X,$ so in particular it is non-zero if and only if $A(X)$ is nonzero. Is there a natural variety assigned to $X$, which is of dimension $h^{n,0}$ (geometric genus $p_g(X)$), or at least a natural geometric object measuring whether $p_g$ is nonzero?