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We generalize the Strichartz estimates for Schr\"odinger operators on compactmanifolds of Burq, G\'erard and Tzvetkov [10] by allowing critically singularpotentials $V$. 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 $V\inL^{1+\delta}(M)$, $\delta>0$, if $n=2$, with $(p,q)$ being as in the Keel-Taotheorem and $I\subset {\mathbb R}$ a bounded interval. We do this byformulating and proving new "quasimode" estimates for scaled dyadic unperturbedSchr\"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 aresaturated on {\em any} compact manifolds; however, we suggest that they maylend themselves to improved Strichartz estimates in certain geometries usingrecently developed "Kakeya-Nikodym" techniques developed to obtain improvedeigenfunction estimates assuming, say, negative curvatures.

Quasimode and Strichartz estimates for time-dependent Schr\"odinger equations with singular potentials