Disclosed is a direct-injection spark-ignition engine designed to promote catalyst activation during cold engine operation. A fuel injection timing for a fuel injection period in an compression stroke (second fuel injection period F2) is set to allow a first fuel spray Ga injected from a first spray hole 40a to enter a cavity 34 in a piston crown surface 30, and allow a second fuel spray Gb to impinge against a region of the piston crown surface 30 located closer to an injector than the cavity 34, so as to cause the second fuel spray Gb having a lowered penetration force due to the impingement to be pulled toward the cavity 34 by a negative pressure generated in the cavity 34 as a result of passing of the first fuel spray Ga therethrough. The direct-injection spark-ignition engine can maximally hold an injected fuel spray around a spark plug to reliably stabilize a combustion state in a combustion mode for promoting catalyst activation during cold engine operation, while enhancing combustion efficiency in a homogenous combustion mode during normal engine operation.