  The two-dimensional quantum $XY$ model with a transverse magnetic field was
investigated with the exact diagonalization method. Upon turning on the
magnetic field $h$ and the $XY$-plane anisotropy $\eta$, there appear a variety
of phase boundaries, which meet at the multicritical point $(h,\eta)=(2,0)$. We
devote ourselves to the Ising-universality branch, placing an emphasis on the
multicritical behavior. As a probe to detect the underlying phase transitions,
we adopt the fidelity susceptibility $\chi_F$. The fidelity susceptibility does
not rely on any presumptions as to the order parameter involved. We made a
finite-size-scaling analysis of $\chi_F$ for $\eta=1$ (Ising limit), where a
number of preceding results are available. Thereby, similar analyses with
$\eta$ scaled were carried out around the multicritical point. We found that
the $\chi_F$ data are described by the crossover scaling theory. A comparison
with the preceding studies of the multicriticality is made.
