Albanese variety ($\mathrm{Alb}(X)$)

The Albanese variety of a smooth projective variety $X$ is the universal abelian variety receiving a morphism from $X$ after a base point is chosen. Over the complex numbers its dimension is $q(X)$.

If the image of the Albanese morphism of a surface is a smooth curve $C$ of genus $g$, then $q(X)=g$. Pullback of one-forms from $C$ gives $g\le q(X)$, while the universal property gives a surjection from the Jacobian of $C$ onto $\mathrm{Alb}(X)$ and hence $q(X)\le g$.