  The honeycomb-lattice Ising antiferromagnet subjected to the imaginary
magnetic field $H=i\theta T /2$ with the "topological" angle $\theta$ and
temperature $T$ was investigated numerically. In order to treat such a
complex-valued statistical weight, we employed the transfer-matrix method. As a
probe to detect the order-disorder phase transition, we resort to an extended
version of the fidelity $F$, which makes sense even for such a non-hermitian
transfer matrix. As a preliminary survey, for an intermediate value of
$\theta$, we investigated the phase transition via the fidelity susceptibility
$\chi_F^{(\theta)}$. The fidelity susceptibility $\chi_F^{(\theta)}$ exhibits a
notable signature for the criticality as compared to the ordinary quantifiers
such as the magnetic susceptibility. Thereby, we analyze the end-point
singularity of the order-disorder phase boundary at $\theta=\pi$. We cast the
$\chi_F^{(\theta)}$ data into the crossover-scaling formula with $\delta \theta
= \pi-\theta$ scaled carefully. Our result for the crossover exponent $\phi$
seems to differ from the mean-field and square-lattice values, suggesting that
the lattice structure renders subtle influences as to the multi-criticality at
$\theta=\pi$.
