nLab moduli space of complex structures

Idea

A moduli space/moduli stack of complex structures on a given topological manifold/smooth manifold.

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.

Related concepts

• Teichmüller theory

• Kodaira-Spencer theory

moduli spaces

• moduli space of complex structures

• moduli space of conformal structures

• moduli space of curves

• moduli space of Riemannian metrics

• moduli stack of elliptic curves

• moduli space of Calabi-Yau spaces

• moduli stack of tori

• moduli stack of formal groups

• moduli space of connections, moduli space of flat connections

• Hitchin moduli space

• moduli space of monopoles

• moduli space of framed manifolds

• moduli space of bundles

• moduli space of (higher) line bundles

• quiver variety

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)