  Wei-Yue Ding \cite{Ding 2002} proposeed a proposition about Schr\"odinger map
flow in 2002 International Congress of Mathematicians in Beijing, which is
called Wei-Yue Ding conjecture by Rodnianski-Rubinstein-Staffilani
\cite{Rodnianski 2009}. They proved \cite{Rodnianski 2009} that Schr\"odinger
map flow for maps from the real line into K\"ahler manifolds and for maps from
the circle into Riemann surfaces is globally well-posed which is the first
significant advance in this conjecture by translating the Schr\"odinger map
flow into nonlinear Schr\"odinger-type equations or (systems) and partially
solved this conjecture. In this article, we will derive a new div-curl type
lemma and combined it with energy and ``momentum" balance law to get some
space-time estimates. Based on this, we prove the Schr\"odinger map flow for
maps from the circle into K\"ahler manifolds is globally regular. So far, the
Wei-Yue Ding's conjecture has been completely solved.
