Contents

complex geometry

# Contents

## Idea

For complex structures on 2-dimensional manifolds the moduli space of complex structures modulo homeomorphisms isotopic to the identity is Teichmüller space, and the further orbifold quotient of that by the mapping class group is an orbifold/Deligne-Mumford stack called the Riemann moduli stack of complex curves.

Specifically for once-punctured 2-dimensional manifolds of genus 1 (tori) this is equivalently the moduli stack of elliptic curves over the complex numbers.

## References

The moduli space of complex structures on a manifold is discussed for instance from page 175 on of

• Yongbin Ruan, Symplectic topology and complex surfaces in Geometry and analysis on complex manifolds (1994)

and in

• Yurii M. Burman, Relative moduli spaces of complex structures: an example (arXiv:math/9903029)

Last revised on October 14, 2019 at 12:47:08. See the history of this page for a list of all contributions to it.