  We study the edge-weighted online stochastic matching problem. Since Feldman,
Mehta, Mirrokni, and Muthukrishnan proposed the $(1-\frac1e)$-competitive
Suggested Matching algorithm, there has been no improvement for the general
edge-weighted online stochastic matching problem. In this paper, we introduce
the first algorithm beating the $1-\frac1e$ barrier in this setting, achieving
a competitive ratio of $0.645$. Under the LP proposed by Jaillet and Lu, we
design an algorithmic preprocessing, dividing all edges into two classes. Then
based on the Suggested Matching algorithm, we adjust the matching strategy to
improve the performance on one class in the early stage and on another class in
the late stage, while keeping the matching events of different edges highly
independent. By balancing them, we finally guarantee the matched probability of
every single edge.
