On smooth affine surfaces with the cohomology of a smooth projective curve

We prove that if a smooth affine complex surface has the same rational mixed Hodge structure as a smooth projective curve of positive genus, then it admits a smooth morphism to the curve with fibers isomorphic to affine line. In particular, every such surface is the total space of a torsor for a line bundle on the curve. The motivation comes from Jouanolou's trick, to which this result gives a kind of converse in dimension 2.