Reference request: bundle of holomorphic differentials over Teichmüller space

Question

Let $S$ be a compact orientable surface of genus $\geq 2$ and let $\mathcal{M}_{-1}(S)$ be the space of smooth hyperbolic metrics on $S$. Then one can define the Teichmüller space on $S$ as $\mathcal{T}(S)=\mathcal{M}_{-1}(S)/\mathrm{Diff}_0(S)$, where $\mathrm{Diff}_0(S)$ denotes diffeomorphisms isotopic to the identity. By the Riemann-Roch theorem, for each $[g]\in\mathcal{T}(S)$, the space of holomorphic $k$-differentials with respect to the complex structure determined by $[g]$ is a finite dimensional vector space. From my understanding, it is known that for each $k$ there exists a finite rank bundle over $\mathcal{T}$ whose fiber at $[g]$ is this space of $k$-holomorphic differentials with respect to the complex structure determined by $[g]$. Is there a good reference for this fact? In particular I am interested in understanding how the trivializations are constructed.

Answer 1

As a first remark, unfortunately I don't know a precise reference for the definition of the spaces $\mathcal Q^k(\mathcal T(S))$ of holomorphic $k$-differentials, for $k\ge 2$. I learned about the construction below just by putting together all my notions on holomorphic vector bundles, the construction of the Teichmüller space as in Tromba's book (see below) and the Riemann-Roch Theorem.

First of all, by the Poincarè-Koebe Uniformization theorem and using that in real dimension $2$ the datum of a conformal class of metrics is equivalent to the datum of a complex structure on the surface, the space $\mathcal M_{-1}(S)$ is in bijection with $\mathcal A$, the space of (almost)-complex structure on $S$. Defining a natural action of $\text{Diff}_0(S)$ on $\mathcal A$, one can prove there is an equivalent description of the Teichmüller space as the quotient $\mathcal A/{\text{Diff}_0(S)}$ (see the wonderful book of A. J. Tromba: "Teichmüller Theory in Riemannian Geometry").

It is a standard result that the Teichmüller space $\mathcal T(S)$ has a smooth manifold structure and it is diffeomorphic to an open ball of dimension $6g-6$. In particular, it is a contractible space. Hence, any vector bundle on it is isomorphic to the trivial vector bundle.

Let $\mathcal Q^k(\mathcal T(S))$ be the total space of the vector bundle of holomorphic $k$-differentials over $\mathcal T(S)$. It can be described as the space of pairs $(J,q)$, where $J$ is an (almost)-complex structure on the surface and $q$ is a $J$-holomorphic section of $K_S^{\otimes^k}$, namely the $k$-tensor product of the canonical bundle $K_S$ with itself. The projection $\pi:\mathcal Q^k(\mathcal T(S))\to\mathcal T(S)$ is simply given by $\pi(J,q)=J$, and the fibre $\pi^{-1}(J)$ over a point $J\in\mathcal T(S)$ is the $\mathbb C$-vector space $H^0(S,K_S^{{\otimes}^k})$. It is a nice exercise, using the Riemann-Roch theorem, to show that the aforementioned vector space is finite dimensional and to compute its complex dimension (for example if $k=3$, then $\text{dim}_{\mathbb C}H^0(S,K_S^{\otimes^3})=5g-5)$.

Answer 2

This is a very good question. I ran into the same issue some months ago, and found it very difficult to find a reference. I warmly recommend Bers' 1960 paper "Holomorphic Differentials as Functions of Moduli."

The starting point is that holomorphic differentials on a Riemann surface are equivalent to holomorphic functions on the universal cover satisfying appropriate transformation properties. Bers, following the procedure behind the Bers emedding, creates a bundle over the Teichmüller space whose fibers are the domains in $\mathbb{CP}^1$ that arise images of the disk under appropriate solutions to Beltrami equations.

The source of trivializations in this perspective is the remarkable Theorem II, which states that every holomorphic differential on a fixed Riemann surface, viewed as a holomorphic function on one fiber of the bundle, is the restriction of a holomorphic global family of differentials on fibers.