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), 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 $\mathbb{C}^3$?
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$\begingroup$ Seems that to get started one would need equations for $\Sigma_g$, see e.g., arxiv.org/abs/math/0605734 then one would have an implicitization problem, see e.g. sciencedirect.com/science/article/pii/S0021869303001819 $\endgroup$– Abdelmalek AbdesselamCommented May 27, 2021 at 16:03
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$\begingroup$ I am not sure I understand the question. As a complex variety, $T^*\Sigma _g$ can certainly not embed into $\mathbb{C}^3$, since it contains a copy of the compact variety $\Sigma _g$. $\endgroup$– abxCommented May 27, 2021 at 16:31
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$\begingroup$ @abx: I think the question is about cotangent bundles with their canonical symplectic forms, for which the zero section is Lagrangian, and trying to find a compatible complex structure which makes the cotangent bundle an affine variety (i.e. a different complex structure to the one you have in mind). $\endgroup$– Jonny EvansCommented May 27, 2021 at 17:05
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$\begingroup$ @Jonny Evans: You are certainly right, thank you. The question might have been more explicit... $\endgroup$– abxCommented May 27, 2021 at 18:15
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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:
answered May 27, 2021 at 17:02
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2$\begingroup$ It was actually proved by Totaro in the early 90s: worldscientific.com/doi/10.1142/S0129167X91000314. The key point is that the Hodge theory of the compactification should contain information of the "growth measure" of its affine piece. $\endgroup$– YHBKJCommented May 27, 2021 at 21:47
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$\begingroup$ @YHBKJ: Thanks! I didn't know this paper. $\endgroup$ Commented May 28, 2021 at 6:06