I am interested in the number of complex structures on a surface. More precisely, given a genus $g$ surface (topological manifold of real dimension 2) with $n$ punctures $X_{(g,n)}$, how many complex structures (up to biholomorphic maps) are there? The usual answer I can find online are for those without punctures.
Also, is there any formula that describes the size of $[X_{(g,n)} , X_{(g',n')}]$, which is defined to be the set of all holomorphic maps (up to biholomorphic maps) $$ \mbox{i.e. } \mbox{Aut}(X_{(g,n)})\backslash \mbox{ Holo}(X_{(g,n)},X_{(g',n')}) \,/ \mbox{Aut}(X_{(g',n')}), $$ from the left to the right?
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4$\begingroup$ Is not $\dim \mathcal{M}_{g, \, n}$ known? In genus $\geq 2$ the general curve has finite automorphism group (and trivial automorphism group for $g \geq 3$) so the dimension is $3g-3+n$. $\endgroup$– Francesco PolizziCommented Nov 20, 2018 at 9:09
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$\begingroup$ I am not familiar with this field.. would you mind pointing out some reference? $\endgroup$– StudentCommented Nov 20, 2018 at 13:56
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1 Answer
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The number is one if $g=0, n\leq 3$ and infinite in all other cases. The set of complex structures is a singular manifold which is called the moduli space and its complex dimension is $1$ when $g=1, n=0$, and $3g-3+n$ in all other cases. A good reference is
W. Abikoff, The real analytic theory of Teichmuller space, Springer 1980.
answered Nov 20, 2018 at 14:09