A synchronous reluctance rotary electric machine of an embodiment includes a shaft and a rotor core. The shaft rotates around a rotation axis. The rotor core is fixed to the shaft, includes four layers of hollow parts having a convex shape toward a radially inner side formed for each pole in cross section, and includes bridges between the respective hollow parts and an outer circumferential surface thereof. Then, in each pole, when a center in a circumferential direction is a pole center, both ends in the circumferential direction are pole ends, and the plurality of bridges are a first layer bridge, a second layer bridge, a third layer bridge, and a fourth layer bridge in order from the pole center toward each of the pole ends, a circumferential width between the second layer bridge and the third layer bridge is set to be greater than a circumferential width between the first layer bridge and the second layer bridge and a circumferential width between the third layer bridge and the fourth layer bridge.