Algebraic geometric model for symplectic $T^* \Sigma_g$?

Question

I was aware of an algebraic geometric model of symplectic $T^* S^2$ recently, that it is $\{x_1^2+x_2^2+x_3^2=1\}$ in $\mathbb{C}^3$, which the Lagrangian $S^2$ is just the real part, and in this way we have 2 foliations of lines. First I had a hard time visualizing this fact even after I saw all the symplectomorphisms written down, and I also wonder whether there are higher genus analogues to this model so that we can see lines or conics or whatever somehow clearly?

Answer 1

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.