| In this thesis, we calculate the Euler characteristics of Teichmuller curves in the moduli space M2 of genus two Riemann surfaces. For any integer D ≡ 0 or 1 (mod 4), there is a Teichmuller curve WD immersed in M2 , which was introduced by McMullen in [47]. The curve WD is naturally embedded in the Hilbert modular surface, HxH/SLOD ⊕O∨D . Our main result is that the Euler characteristic of WD is proportional to the Euler characteristic of XD. More precisely, cWD= -92cXD , when D is not square.; When D ≡ 1 (mod 8), the curve WD has two connected components, W0D and W1D . We also calculate the Euler characteristics of these components. When D is not square, we show that cW0D =cW1D . When D is square, we show that cW0d2 =-132d2 d-1r&vbm0; dmr r2,and cW1d2 =-132d2 d-3r&vbm0; dmr r2.; The idea of the calculation of chi(WD) is to use techniques from algebraic geometry to compute the fundamental class of the closure WD in a compactification of XD. We define a compactification YD of XD which maps to the Deligne-Mumford compactification of M2 by a finite morphism. We then exhibit WD as the zero locus of a meromorphic section of a line bundle over YD, which allows us to calculate the fundamental class of WD.; To calculate the Euler characteristics of the connected components WeD , we find several relations involving the fundamental class of the closure of WeD which allow us to solve for chi( WeD ). For example, we calculate the self-intersection numbers of the WeD in terms of chi( WeD ). |
