  The variations of the 3-D coronal magnetic fields associated with the
X3.4-class flare of active region 10930 are studied in this paper. The coronal
magnetic field data are reconstructed from the photospheric vector magnetograms
obtained by the Hinode satellite and using the nonlinear force-free field
extrapolation method developed in our previous work (He et al., 2011). The 3-D
force-free factor $\alpha$, 3-D current density, and 3-D magnetic energy
density are employed to analyze the coronal data. The distributions of $\alpha$
and current density reveal a prominent magnetic connectivity with strong
negative $\alpha$ values and strong current density before the flare. This
magnetic connectivity extends along the main polarity inversion line and is
found to be totally broken after the flare. The distribution variation of
magnetic energy density reveals the redistribution of magnetic energy before
and after the flare. In the lower space of the modeling volume the increase of
magnetic energy dominates, and in the higher space the decrease of energy
dominates. The comparison with the flare onset imaging observation exhibits
that the breaking site of the magnetic connectivity and site with the highest
values of energy density increase coincide with the location of flare initial
eruption. We conclude that a cramped positive $\alpha$ region appearing in the
photosphere causes the breaking of the magnetic connectivity. A scenario for
flare initial eruption is proposed in which the Lorentz force acting on the
isolated electric current at the magnetic connectivity breaking site lifts the
associated plasmas and causes the initial ejection.
