  The effect of quark anomalous magnetic moment (AMM) to chiral restoration and
deconfinement phase transitions under magnetic fields is investigated in a
Pauli-Villars regularized PNJL model. A linear-in-$B$ term for quark anomalous
magnetic moment is introduced to the Lagrangian density of our model, and it
plays the role of inverse catalysis to the phase transitions. With fixed
magnetic field, the critical temperature decreases with quark AMM. When fixing
quark AMM, the critical temperature increases with magnetic field for a small
quark AMM, but decreases with magnetic field for a large quark AMM. The
critical temperature of chiral restoration and deconfinement phase transitions
is determined by the two competing factors, the catalysis effect of magnetic
field and inverse catalysis of quark anomalous magnetic moment.
