  We generalize the Strichartz estimates for Schr\"odinger operators on compact
manifolds of Burq, G\'erard and Tzvetkov [10] by allowing critically singular
potentials $V$. Specifically, we show that their $1/p$--loss
$L^p_tL^q_x(I\times M)$-Strichartz estimates hold for $e^{-itH_V}$ when
$H_V=-\Delta_g+V(x)$ with $V\in L^{n/2}(M)$ if $n\ge3$ or $V\in
L^{1+\delta}(M)$, $\delta>0$, if $n=2$, with $(p,q)$ being as in the Keel-Tao
theorem and $I\subset {\mathbb R}$ a bounded interval. We do this by
formulating and proving new "quasimode" estimates for scaled dyadic unperturbed
Schr\"odinger operators and taking advantage of the the fact that
$1/q'-1/q=2/n$ for the endpoint Strichartz estimates when $(p,q)=(2,2n/(n-2))$.
We also show that the universal quasimode estimates that we obtain are
saturated on {\em any} compact manifolds; however, we suggest that they may
lend themselves to improved Strichartz estimates in certain geometries using
recently developed "Kakeya-Nikodym" techniques developed to obtain improved
eigenfunction estimates assuming, say, negative curvatures.
