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Given an algebraic torus $(\mathbb{C}^\ast)^n$, what's the moduli space of complex structures? Even for $\mathbb{C}^\ast$, since it's a non-compact Riemann surface with puncture, it doesn't seem trivial for me. Does anyone know some references on this?

Many toric varieties don't have moduli, e.g. $\mathbb{CP}^2$, but in the case of $(\mathbb{C}^\ast)^n$, the complex structure doesn't need to extend to the toric boundaries. This consideration implies that $(\mathbb{C}^\ast)^n$ may have larger moduli than $\mathbb{C}^n$.

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  • $\begingroup$ What is your definition of "moduli of complex structures" for a non-compact complex manifold? For instance, if you take the complex manifold $M=\mathbb{C}^2 \setminus\{(z,w) : |z|^2 \leq |w|^2\}$ with its standard projection onto the $w$-line, do you consider that a "deformation" of the fiber $\mathbb{C}^*$ over $w=0$? $\endgroup$
    – Jason Starr
    Commented Nov 15, 2013 at 13:00
  • $\begingroup$ Correction: "projection onto the $w$-line" --> "projection onto the $z$-line"; "$w=0$" --> "$z=0$". $\endgroup$
    – Jason Starr
    Commented Nov 15, 2013 at 13:38

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There is just no definition of the moduli for complex structures on non-compact manifolds, but by any reasonable definition, it would be (generally) very bad space, certainly infinite-dimensional. For example, if you are interested in the bounded Stein subsets of ${\Bbb C}^n$ with smooth boundary, the boundary (more precisely, a codimension 1 subbundle of its tangent bundle) acquires a semi-definite Hermitian form, called the Levi form. This form is an invariant of a complex structure, which can be chosen pretty much in any way. Therefore, the space of such manifolds has more or less the same size as the space of K\"ahler forms (which is more or less same size as the space of smooth functions). The moduli of algebraic structures are probably even worse, because a given complex (even Stein) manifold can support many compatible algebraic structures (probably an infinite-dimensional families of algebraic structures, though I don't have a proof handy).

Most likely this can be applied to $({\Bbb C}^*)^n$: this space is Stein and it certainly has at least two non-equivalent algebraic structures.

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