Let $X$ be a compact complex manifold. People always refer the Kuranishi family of $X$ as the "local moduli space" of the moduli space of complex structures on $X$.

I am confused with this terminology. When I look at the 2-torus case $T^2$. It is known that the moduli space of complex structure on $T^2$ is given by the quotient $$\mathbb{H}/SL(2,\mathbb{Z}),$$ where $\mathbb{H}$ is the upper-half space and $SL(2,\mathbb{Z})$ acts on it by Mobius transform. However, the Kuranishi space of $T^2$ is simply given by the unit disk (or $\mathbb{H}$), because the Maurer-Cartan element is given by $$td\bar{z}\otimes\frac{\partial}{\partial z}.$$ Hence for different $t$'s, the corresponding elliptic curves can be isomorphic. So, how should I interpret the Kuranishi family as the "local moduli space"?

Thank you!

  • $\begingroup$ I'm sorry I can't help much with this question (I don't know some of the concepts you're using), but is it possible that the Kuranishi family of $X$ is actually the Teichmuller space of $X$? The moduli space is a quotient of the Teichmuller space. For a little information on this, see this delightful paper. $\endgroup$ – Myridium Oct 17 '16 at 7:29
  • $\begingroup$ Never mind. At least your comment leads me to read the following paper arxiv.org/pdf/1106.1368.pdf. Thank you anyway! $\endgroup$ – Marco Oct 17 '16 at 7:45

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