I was reading about the blockchain, as I wanted to make a small implementation. What I did not understand is what happens if a miner adds valid PoW to a block with transactions that have an invalid digital signature, why doesn't the blockchain just continue with the forged block and keep stacking blocks on top of it? How is it "corrected"?
The behavior of miners and other nodes is determined by the specific protocol of the network. That is, either such a block can be considered invalid and not accepted as "next", or the block will be accepted, but the transactions included in it will be ignored. In some blockchains, the miner who created such a block can be somehow fined - excluded from the pool of miners, deprived of part of the collateral.