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In this paper and the companion work \cite{LIZE2}, we prove that theSchr\"odinger map flows from $\Bbb R^d$ with $d\ge 2$ to compact K\"ahlermanifolds with small initial data in critical Sobolev spaces are global. Themain difficulty compared with the constant sectional curvature case is that thegauged equation now is not self-contained due to the curvature part. Our mainidea is to use a novel bootstrap-iteration scheme to reduce the gauged equationto an approximate constant curvature system in finite times of iteration. Thispaper with the companion work \cite{LIZE2} solves the open problem raised byTataru.

Global Schr\"odinger map flows to K\"ahler manifolds with small data in critical Sobolev spaces: Energy critical case