As far as I understand, it is easy to see (and find in the literature) that the affine variety
$$z\_1^2+z\_2^2+z\_3^2=1$$
with the restriction of the standard $\omega\_{std}$ of $\mathbb{C}^3$ is symplectomorphic to $T^\*S^2$ with the standard complex structure. Now, as mentioned here ([https://www.math.stonybrook.edu/~markmclean/talks/cotangentaffine.pdf](https://www.math.stonybrook.edu/%7Emarkmclean/talks/cotangentaffine.pdf)), also $T^\*T^2$ can also be written explicitly as an affine variety. So my question is: Is there a general way to write $T^\*\Sigma\_g$ as the zero set of some polynomial in $\mathbb{C}^3$?



Cotangent bundles of surfaces as varieties
This is not possible. There is something called the growth rate of symplectic cohomology which is subexponential for affine varieties and exponential for cotangent bundles of higher genus surfaces (amongst other things). This was proved by McLean:


<https://arxiv.org/abs/1011.2542>


