ag.algebraic geometry - Algebraic geometric model for symplectic $T^* Sigma_g$?

The cotangent bundle to $T^2$ is $(mathbb{C}^ast)^2$. But a theorem of Totaro (Internat. J. Math. 2 (1991), 563-566; MR1124283) implies that, if $S$ is a closed orientable surface of negative Euler characteristic, there is no diffeomorphism, sending the symplectic orientation to the complex orientation, from $T^*S$ to a smooth affine complex surface.