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A toy SYZ mirror symmetry is described as follows. Let $X$ be a $n$-dimensional compact manifold. The contangent bundle $T^*X$ has a natural symplectic structure locally given by the $2$-form $\sum_idq_i \wedge dx_i$. Here $(x_i)$ is a local coordinate of $X$ and $(x_i,q_i)$ is a local coordinate of $T^*X$ corresponding to the $1$-form $\sum_i q_i dx_i$.

On the other hand, the tangent bundle $TX$ has a complex structure locally given by $(x_i+iy_i)$. Here $(x_i)$ is a local coordinate of $X$ and $(x_i,y_i)$ is a local coordinate of $TX$ corresponding to the vector field $\sum_i y_i \partial/\partial x_i$.

Some people want to compactify the two spaces by modding out the fibers by a lattice and its dual. Then the local holomorphic coordinate becomes $z_i=\exp2\pi \sqrt{-1}(x_i+iy_i)$. This is a toy SYZ mirror symmetry I am aware of.

My question is, why the complex structure on $TX$ is natural? Alsoare there any more good examples that shows the SYZ mirror symmetry?

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In "Mirror symmetry and T-duality in the complement of an anticanonical divisor", Denis Auroux works a number of examples of SYZ for Fano varieties. The toric Fano case is a good starting point: everything is totally concrete and geometric. (An unrelated reason to read that paper is that it's the only place I know of where the folk theorem that the self-Floer cohomology of a Lagrangian is a module over quantum cohomology is written down.) – Nathaniel Bottman Apr 12 '13 at 15:54
@Simon: You seem to be operating under the mistaken impression that there is a natural complex structure on $TX$. The expression for a complex structure in terms of local coordinates that you give will not patch when you make an arbitrary change coordinates, so you have not really defined a complex structure on $TX$ by this method. More generally, one can prove that there is no way to assign a complex structure $J_X$ to $TX$ for all $n$-manifolds $X$ in such a way that every diffeomorphism $f:X\to Y$ induces a biholomorphism $f':TX\to TY$. – Robert Bryant Apr 13 '13 at 13:31
As Robert pointed, there is in general no natural way to assign a complex structure on $TX$. However, when $X$ is an affine manifold, this is possible. – Atsushi Kanazawa Apr 14 '13 at 6:09

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