  In this paper, we prove that the Schr\"odinger map flows from $\Bbb R^d$ with
$d\ge 3$ to compact K\"ahler manifolds with small initial data in critical
Sobolev spaces are global. This is a companion work of our previous paper [23]
where the energy critical case $d=2$ was solved. In the first part of this
paper, for heat flows from $\Bbb R^d$ ($d\ge 3$) to Riemannian manifolds with
small data in critical Sobolev spaces, we prove the decay estimates of moving
frame dependent quantities in the caloric gauge setting, which is of
independent interest and may be applied to other problems. In the second part,
with a key bootstrap-iteration scheme in our previous work [23], we apply these
decay estimates to the study of Schr\"odinger map flows by choosing caloric
gauge. This work with our previous work solves the open problem raised by
Tataru.
