nLab moduli space of curves

Idea

The moduli space/moduli stack of algebraic curves/Riemann surfaces is a sort of space of parameters parametrizing algebraic curves of given genus.

Deligne-Mumford 69 a nontrivial compactification of a moduli space of Riemann surfaces of fixed genus which is a Deligne-Mumford stack, since called the Deligne-Mumford compactification.

There is also a decorated version of curves with marked points, and of the corresponding compactified moduli space of stable curves of genus $g$ with $n$ marked points $\mathcal{M}_{g,n}$ which plays an important role in the mathematical study of Gromov-Witten invariants and of conformal blocks.

The special case of $g = 1$, $n =1$ is the moduli stack of elliptic curves $\mathcal{M}_{1,1}= \mathcal{M}_{ell}$.

Properties

Over the complex numbers

Remark

On 2-dimensional manifold every almost complex structure is integrable and hence is already a complex structure (see this proposition).

Hence the moduli space of complex surfaces is also equivalent to just the quotient orbifold of the space of almost complex structures by the orientation-preserving part of the diffeomorphism group.

This is reviewed for instance in (Madsen 07, section 1.1).

Remark

We indicate how via remark the moduli space of complex curves has a formulation in the homotopy-type theory $\mathbf{H}$ of smooth homotopy types.

By the discussion at almost complex structure (see this remark), if the tangent bundle of a $2n$-dimensional smooth manifold is modulated by a map

$$\tau_X \;\colon\; X \longrightarrow \mathbf{B}GL(2n,\mathbb{R})$$

to the delooping in smooth stacks of the general linear group, then an almost complex structure on $X$ is equivalently a lift $J$ in

$$\array{ X && \stackrel{J}{\longrightarrow} && \mathbf{B} GL(n,\mathbb{C}) \\ & {}_{\mathllap{\tau}}\searrow && \swarrow_{\mathrlap{almComp}} \\ && \mathbf{B} GL(2n,\mathbb{R}) } \,.$$

This in turn is equivalently a map

$$J \;\colon\; \tau_X \longrightarrow almComp$$

in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}GL(2n,\mathbb{R})}$, hence with the canonical empedding

$$\tau_{(-)} \;\colon\; SmthMfd_{2n}^{et} \hookrightarrow \mathbf{H}_{/\mathbf{B}GL(2n,\mathbb{R})}$$

of the category of smooth manifolds with local diffeomorphisms between them understood, then $almComp \in \mathbf{H}_{/\mathbf{B}GL(2n,\mathbb{R})}$ is the universal moduli stack of almost complex structures.

Now $\tau_X$ carries a canonical ∞-action by the diffeomorphism group. Using this one may canonically form the homotopy quotient

$$[\tau_X, almComp]//Diff(X)$$

by a general abstract construction that is discussed in some detail at general covariant – Formalization in homotopy type theory. For $n =1$ this is hence the Riemann moduli space.

Homotopy type and cohomology

The homotopy type of the Riemann moduli spaces $\mathcal{M}_g$ (i.e. of the geometric realization of the orbifold, hence essentially of the delooping of the mapping class group) is essentially unknown. Even its orbifold cohomology over the rational numbers is fully known only for $g \leq 4$.

However, in a stable range it has been fully computed in (Madsen-Weiss 02), proving what was previously conjectured by Mumford's conjecture. A review is in (Madsen 07).

Orbifold Euler characteristic

The orbifold Euler characteristic $\chi$ of $\mathcal{M}_{g,1}$ is given by the Riemann zeta function at negative integral values as follows (Zagier-Harer 86):

$$\chi(\mathcal{M}_{g,1}) = \zeta(1-2g) \,.$$

By the expression of the Riemann zeta function at negative integral values by the Bernoulli numbers $B_n$, this says equivalently that

$$\chi(\mathcal{M}_{g,1}) = -\frac{B_{2g}}{2g} \,.$$

For instance for $g = 1$ (once punctured complex tori, hence complex elliptic curves) this yields

$$\chi(\mathcal{M}_{1,1}) = -\frac{1}{12}$$

for the orbifold Euler characteristic of the moduli space of elliptic curves.

Relation to Teichmüller space

Over the complex number then the moduli space of curves is a quotient of Teichmüller space. (Hubbard-Koch 13)

Examples

$g = 0$, $n = 0$

By the Riemann mapping theorem?, $\mathcal{M}_{0,0}\simeq \ast$ is the point.

$g = 1$, $n = 0$

$\mathcal{M}_{1,0} \simeq \mathbb{R}^2$

$g = 1$, $n = 1$

$\mathcal{M}_{1,1} = \mathcal{M}_{ell}$, the moduli stack of elliptic curves.

$g \geq 2$, $n = 0$

For genus $g\geq 2$ the moduli stack of complex structures is equivalently that of hyperbolic metrics. This way a lot of hyperbolic geometry is used in the study of $\mathcal{M}_{g \geq 2, n}$.

Related entries

• moduli space, moduli stack, mapping class group, enumerative geometry, Mumford class, Deligne-Mumford stack

• moduli space of super Riemann surfaces

• moduli space of (higher) line bundles

• Witten conjecture

References

Original articles:

• Pierre Deligne, David Mumford, The irreducibility of the space of curves of given genus, Publications Mathématiques de l’IHÉS (Paris) 36: 75–109 (1969) numdam

• David Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328, MR85j:14046

On moduli spaces of curves/moduli spaces of Riemann surfaces with emphasis on their orbifold-structure:

• Lizhen Ji, Shing-Tung Yau, Transformation Groups and Moduli Spaces of Curves, International Press of Boston 2011 (ISBN:9781571462237)

Discussion of the orbifold cohomology of the moduli stack is in

• John Harer, The cohomology of the moduli space of curves, Lec. Notes in Math. 1337, p. 138–221. Springer, Berlin, 1988.

The stable rational orbifold cohomology of the moduli stack was computed in

• Ib Madsen, Michael Weiss, The stable moduli space of Riemann surfaces: Mumford's conjecture, Ann. of Math. (2) 165 (2007), no. 3, 843–941, MR2009b:14051, doi, math.AT/0212321

exposition of this is in

• Ib Madsen, Moduli spaces from a topological viewpoint, Proceedings of the International Congress of Mathematics, Madrid 2006 (2007) (pdf)

See also

• Gabriele Mondello, Combinatorial classes on $\mathcal{M}_{g,n}$ are tautological, Int. Math. Res. Not. 44 (2004), 2329-–2390, MR2005g:14056, doi, math.AG/0303207

• Gabriele Mondello, Riemann surfaces, ribbon graphs and combinatorial classes, in: Handbook of Teichmüller theory. Vol. II, 151–215, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009; draft with index: pdf, arxiv version math.AG/0705.1792, MR2010f:32012

• Alastair Hamilton, Classes on compactifications of the moduli space of curves through solutions to the quantum master equation, Lett. Math. Phys. 89 (2009), no. 2, 115–130.

• The moduli of curves pdf

The orbifold Euler characteristic of the moduli space of curves was originally computed in

• Don Zagier, John Harer, The Euler characteristic of the moduli space of curves, Inventiones mathematicae (1986) Volume: 85, page 457-486 (EUDML)

The complex analytic structure and the relation to Teichmüller space is further discussed in

• John Hubbard, Sarah Koch, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves (arXiv:1301.0062)

Reviews include

• Mathematical ideas and notions in quantum field theory – 5. The Euler characteristic of the moduli space of curves (pdf)

Étale homotopy type of moduli stacks of curves is discussed in

• Paola Frediani, Frank Neumann, Étale homotopy types of moduli stacks of algebraic curves with symmetries, K-Theory 30: 315-340, 2003 (arXiv:math/0404387)