  The transverse-field $XY$ spin chain with competing antiferromagnetic
long-range interactions, $J_r \propto 1/r^\alpha$ ($r$: distance between
spins), and the exponent $\alpha$ was investigated numerically. The main
concern is to clarify the character of the transverse-field-driven phase
transition for the small-$\alpha$ regime around the $XX$-symmetric point,
$\eta=0$ ($\eta$: $XY$-anisotropy parameter). To cope with the negative-sign
problem, we employed the exact-diagonalization method, which enables us to
evaluate the fidelity susceptibility $\chi_F$. Because the fidelity
susceptibility does not assume any order parameter after a phase transition, it
detects the multi-criticality around the $XX$-symmetric point in a systematic
manner. As a preliminary survey, with $\eta=0.5$ and $\alpha=2$ fixed, we made
the scaling analysis of $\chi_F$. The scaling behavior of $\chi_F$ shows that
the transverse-field-driven phase transition belongs to the 2D-Ising
universality class. Thereby, with the properly crossover-scaled $\eta$, the
$\chi_F$ data is cast into the crossover scaling formula for the small-$\alpha$
regime. The multi-critical exponents fed into the scaling formula are argued
through referring to related studies.
