Spark ignition direct injection engine with supercharger

A spark-ignition type 4-cycle direct injection engine is provided with a turbocharger for boosting intake air and an injector for directly injecting fuel to a combustion chamber within a cylinder, wherein in a lambda =1 region (II) and an enriched region (III) on the high-speed high-load side, fuel is injected during the intake stroke of the cylinder to attain a state of homogenous combustion. When the engine is in the high-speed side (specific region) of the enriched region (III), the air-fuel ratio A/F of the air-fuel mixture in the cylinder is controlled to become A/F