Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.