On the Weil-Petersson metric on Teichmüller space

Teichmüller space for a compact oriented surface $M$ without boundary is described as the quotient $\mathcal {A}/{\mathcal {D}_0}$, where $\mathcal {A}$ is the space of almost complex structures on $M$ (compatible with a given orientation) and ${\mathcal {D}_0}$ are those ${C^\infty }$ diffeomorphisms homotopic to the identity. There is a natural ${\mathcal {D}_0}$ invariant ${L_2}$ Riemannian structure on $\mathcal {A}$ which induces a Riemannian structure on $\mathcal {A}/{\mathcal {D}_0}$. Infinitesimally this is the bilinear pairing suggested by Andre Weil—the Weil-Petersson Riemannian structure. The structure is shown to be Kähler with respect to a naturally induced complex structure on $\mathcal {A}/{\mathcal {D}_0}$.