  We calculate the form factors for the $B \to D^*\ell\nu_\ell$ decay in 2+1
flavor lattice QCD. For all quark flavors, we employ the M\"obius domain-wall
action, which preserves chiral symmetry to a good precision. Our gauge
ensembles are generated at three lattice cutoffs $a^{-1} \sim 2.5$, 3.6 and 4.5
GeV with pion masses as low as $M_\pi \sim 230$ MeV. The physical lattice size
$L$ satisfies the condition $M_\pi L \geq 4$ to control finite volume effects
(FVEs), while we simulate a smaller size at the smallest $M_\pi$ to directly
examine FVEs. The bottom quark masses are chosen in a range from the physical
charm quark mass to $0.7 a^{-1}$ to control discretization effects. We
extrapolate the form factors to the continuum limit and physical quark masses
based on heavy meson chiral perturbation theory at next-to-leading order. Then
the recoil parameter dependence is parametrized using a model independent form
leading to our estimate of the decay rate ratio between the tau ($\ell = \tau$)
and light lepton ($\ell = e,\mu$) channels $R(D^*) = 0.252(22)$ in the Standard
Model. A simultaneous fit with recent data from the Belle experiment yields
$|V_{cb}| = 39.19(91)\times 10^{-3}$, which is consistent with previous
exclusive determinations, and shows good consistency in the kinematical
distribution of the differential decay rate between the lattice and
experimental data.
