We calculate the Euler characteristics of all of the Teichmüller curves in the moduli
space of genus two Riemann surfaces which are generated by holomorphic one-forms
with a single double zero. These curves are naturally embedded in Hilbert modular
surfaces and our main result is that the Euler characteristic of a Teichmüller curve
is proportional to the Euler characteristic of the Hilbert modular surface on which it
lies.
The idea is to use techniques from algebraic geometry to calculate the
fundamental classes of these Teichmüller curves in certain compactifications of the
Hilbert modular surfaces. This is done by defining meromorphic sections of line
bundles over Hilbert modular surfaces which vanish along these Teichmüller
curves.
We apply these results to calculate the Siegel–Veech constants for counting
closed billiards paths in certain L-shaped polygons. We also calculate the
Lyapunov exponents of the Kontsevich–Zorich cocycle for any ergodic,
–invariant
measure on the moduli space of Abelian differentials in genus two (previously
calculated in unpublished work of Kontsevich and Zorich).
