Bohan Fang

BICMR, Peking University

This is a not very detailed record on what happened in each course meeting. There should be course recordings in 北大教学网.

Sep 08

We discussed the definition of Teichmuller space, and the differentiable structure of $\mathrm{Teich}(T^2)$.

Reference: [FM], pages 263-269.

Sep 10

We discussed the algebraic topology of $\mathrm{Teich}(S_g)$, by identifying it with $\mathrm{DF}(\pi_1(S_g),PSL(2,\mathbb R))/PGL(2,\mathbb R)$.

Then we show that $\mathrm{Teich}(S_0^3)\cong\mathbb R^3_+$, by considering the lengths of the three boundaries.

Any closed surface can be decomposed into pairs of pants. We defined twisting parameters using seams.

Reference: [FM], pages 269-282.

Sep 15

We discussed Fenchel-Nielson coordinates for Teichmuller space, and in particular, showed that $\mathrm{Teich}(S_g)\cong \mathbb R^{3g-3}_+\times \mathbb R^{3g-3}$.

Teichmuller space can be regarded as the space of marked hyperbolic surfaces, or as the space of marked complex structures. We introduced quasi-conformal maps, and stated Teichmuller’s result on the Teichmuller metric using quasi-conformal maps.

Reference: [FM], pages 282-287, 294-300.

Sep 22

We discussed measured foliations and quadratic differentials $\mathrm{QD}(X)$ on a Riemann surface, possibly bordered and punctured. These two concepts are closed related and essentially equivalent. We defined Teichmuller mappings based on the definition of quadratic differentials. We also computed $\mathrm{dim}_{\mathbb C} \mathrm{QD}(X)=3g-3$.

Reference: [FM], pages 300-305, 309-312.

Sep 24

As an example, we studied the Swiss Cross as a construction of a genus $2$ Riemann surface together with a quadratic differential. We stated the Teichmuller existence and uniquess theorem, and explained the diffeomorphism map between $\mathrm{QD}_1(X)$ and $\mathrm{Teich}(S)$. The Teichmuller metric is a complete metric defined by the dilatation of the Teichmuller map relating two conformal structures.

Reference: [FM], pages 313-315, 320-323, 330-332, 337-341.

Oct 15

We introducted mapping class groups, and discussed several examples (disk, annulus, torus with zero or $1$ marked points, $S_{0,3}$ and $S_{0,4}$.) We introduced the concept of the moduli space of (smooth) conformal/hyperbolic structures, and discussed the example of $\mathcal M(T^2)$.

Reference: [FM], Chapter 2, pages 342-349.

Oct 19

We discussed Beltrami differentials and the Betlrami equation. In particular, by solving the Beltrami equation and restricted to harmonic Beltrami differentials, we get the tangent space of the Teichmuller space. We further discussed the complex structure on $\mathcal T_{g,n}$.

The quadratic differential is naturally identified with the cotangent space os the Teichmuller space. We discussed Weil-Pertersson metric as a hermitial metric given by the $L^2$ norm on Beltrami differentials. The Teichmuller metric is given by $L^\infty$-norm, or equivalently the $L^1$-norm on quadratic differentials discussed earlier.

References: [Hu] Ch 4 and 6, [IT] Ch 6. We did a very superficial survey on the deep analysis in this topic.

Oct 22

The moduli space $\mathcal M_{g,n}=\mathcal T_{g,n}/\mathrm{PMod}_{g,n}$ is a complex orbifold from the properly discontinuous action.

We discussed the compactification of the moduli space: first we constructed a partial completion $\hat{\mathcal T}_{g,n}$ by allowing degeneration into a node, together with an extension of the action of the mapping class group.

The Deligne-Mumford compactification is given by the quotient $\overline{\mathcal M}{g,n}=\hat{\mathcal T}{g,n}/\mathrm{PMod}_{g,n}$.

The construction is a priori not a compact space, or an orbifold. The compactness argument follows Mumford’s compactness theorem, while the the orbifold structure at the boundary strata is given locally by the $xy=\epsilon$ near $\epsilon=0$: the boundary stratum is a normal crossing divisor.

References: Orbifold structure on $\overline{\mathcal M}_{g,n}$ can be found in [HV] (a very modern treatment). The compactness discussion is in [FM], Section 12.4.

Oct 28

We reviewed some basic notion of orbifolds, and discussed the universal property of moduli spaces. As an example, $\mathcal M_{1,1}$ has an orbifold universal curve of (smooth) genus $1$ curves with one marked points, but as a scheme (or topological space) the moduli space $M_{1,1}$ is too coarse.

The moduli space $\overline{\mathcal M}_{g,n}$ as an orbifold has a universal curve.

We discussed the example of $\overline{\mathcal M}_{0,4}$ and its universal curve.

The boundary of $\overline{\mathcal M}_{g,n}$ are stratified: they can be described by the dual graph of the topological type of the stable curves.

References: [Zv], pages 671-681.

Nov 3

The universal curve $\overline{\mathcal C}{g,n}$ is isomorphic to $\overline{\mathcal M}{g,n+1}$. We introduced the concept of tautological ring, relative cotangent bundle on the universal curve, Hodge bundle and tautological line bundles.

References: [Zv], pages 673-674, 682-686.

Nov 12

We reviewed the basics of intersection theory on orbifolds, and discussed certain intersection numbers of $\psi$-classes on $\overline{\mathcal M}{g,n}$. In particular, we have shown $$ \int{\overline{\mathcal M}_{1,1}} \psi_1=\frac{1}{24}. $$

References: [Zv], pages 686-690.

Nov 17

Nov 19

Dec 1

Dec 3

  • [FM] Farb, B., & Margalit, D. (2012), A Primer on Mapping Class Groups. Princeton University Press.

  • [HM] Harris, J., & Morrison, I., Moduli of Curves. Springer

  • [HV] Hinich, V., & Vaintrob, A., Augmented Teichmuller spaces and Orbifolds. Selecta Math. (N.S.)

  • [Hu] Hubbard, J. H., Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1.

  • [IT] Imayoshi, Y., Taniguchi, M., An introduction to Teichmuller space

  • [Zv] Zvonkine, D., An introduction to moduli spaces of curves and their intersection theory (EMS published book Chapter 11, in Handbook of Teichmuller Theory vol III).