Which defense variation is this?

This is the Caro-Kann defense, but does anybody know what the variation is called?

Giving up material in exchange for development and two half-open files.

It's called black gives up a pawn for one tempo in compensation gambit.  So in other words, dubious.

So, no official name for this variation? It seems fine for black.

It gives black one tempo for a pawn.  You need three tempo to justify a pawn investment.  So really you will have barely enough compensation to last until white castles to which black will just be fighting in a pawn down middlegame.

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This particular variation does not have a name. Since its only a few moves into the opening you have to figure it's not a great move. But you can also turn on Stockfish to help evaluate the position.

Chessbase gives it a B01 code since it can transpose out of the Scandinavian defense as well.

If you want to give up a pawn, this is way better.

The begin of your line was a Caro-Kann.

However, as soon as you gave up 1 pawn.

You no longer was in a Caro-Kann.

For the Caro-Kann is not a gambit line.

The closest name I can find which might be similar to your position is called the Scandinavian Defense/Blackburne Gambit

The above line is the closest I could get to a name.