I have a hard time deciding when to push that pawn in these kinds of openings.
You get a space advantage
Space, control d3/b3 squares, and fix the pawn on b2.
Just a point of view:
On move #9, White could trade pawns but it would assist Black in developing the Bishop and be better for Black due to 9. dxc4 Bxc4. So instead, White castles with 9. O-O.
On move #9, Black develops the Knight with 9... Ne7 which blocks the Bishop from defending the c4 pawn. From one perspective, Black will lose material, but it will only be a pawn while Black develops their Knight and it will cause White to lose their long pawn chain. However, White will be able to build a new strong pawn formation with 9. O-O Ne7 10. dxc5 ... 11. b4. Even if Black attempts to prevent 11. b4 with 10... a5, it will still allow White to maintain a pawn chain with 9. O-O Ne7 10. dxc5 a5 11. b4 axb4 12. cxb4. In both cases, White is stampeding forward on the Queen side with pawns which is a good attack strategy since Black won't be able to castle kingside for at least two more moves assuming no additional threats are made.
Why is 9... c4 so good for Black? It eliminates the threat of 10. dxc5 while both enabling Bd3 and/or cxb3 should White ever push the b2 pawn.
Stockfish 2021-03-27 Multi-Variant prefers 9... c4 at least to Depth 41. I don't think 9... c4 is absolutely necessary, but 9... Ne7 created problems for Black. 9. O-O c4 10. ... Ne7 would avoid those problems.
On move #9, White could trade pawns but it would assist Black in developing the Bishop and be better for Black due to 9. dxc4 Bxc4. So instead, White castles with 9. O-O.
On move #9, Black develops the Knight with 9... Ne7 which blocks the Bishop from defending the c4 pawn. From one perspective, Black will lose material, but it will only be a pawn while Black develops their Knight and it will cause White to lose their long pawn chain. However, White will be able to build a new strong pawn formation with 9. O-O Ne7 10. dxc5 ... 11. b4. Even if Black attempts to prevent 11. b4 with 10... a5, it will still allow White to maintain a pawn chain with 9. O-O Ne7 10. dxc5 a5 11. b4 axb4 12. cxb4. In both cases, White is stampeding forward on the Queen side with pawns which is a good attack strategy since Black won't be able to castle kingside for at least two more moves assuming no additional threats are made.
Why is 9... c4 so good for Black? It eliminates the threat of 10. dxc5 while both enabling Bd3 and/or cxb3 should White ever push the b2 pawn.
Stockfish 2021-03-27 Multi-Variant prefers 9... c4 at least to Depth 41. I don't think 9... c4 is absolutely necessary, but 9... Ne7 created problems for Black. 9. O-O c4 10. ... Ne7 would avoid those problems.
@chessfan124 it often has to do with who will be favored. By pushing the pawn you are releasing the pressure on white's center. The question is, who will be favored by this. Will it be white, because they don't have to defend their center so much anymore, or will it be black, because they gain space and perhaps make pawn advances harder for white.
In this position, notice two things. White no longer has the light bishop, and white has castled short. Suppose that white had not done either of those, either had the light bishop or had castled long. Then the move c4 would not be good.
Why?
Here is the reason. By playing c4 black is doing several things. First he is gaining a little bit of space, but also he is controlling the d3 square. This is a hole in the white position, and since white does not have the light bishop, a piece of black's can one day use this square potentially. This is especially true because black still has the light bishop. It also extends the scope of the black dark bishop, something which may one day be useful if black goes for a queenside pawn attack. Finally, it denies the white knight on f3 of a square it would love to use, d4. d4 will now be forever home to a white pawn, and the knight on f3 is just that much worse.
Also, c4 is closing down the queenside, something that black is ok with because the white king is not on that side of the board. The action will be on the kingside, and black may continue with moves like Nd7 to f6, Be7, castles, and then the standard f6 break, opening the f-file and playing from there.
If you are interested in looking into this position further, I would encourage you to remove the knight on b1 and replace it with a bishop on e2. Then see what stockfish has to say about c4. I wouldn't be surprised if it doesn't like it anymore.
Be curious about the game, check out some different lines, and the meaning of the position will start to unfold. It helps to thoroughly understand one position, and thus understand many others that are similar.
there are other positions in the french where c4 loses for black. It depends on a lot of factors, but I think I touched on the big ones.
In this position, notice two things. White no longer has the light bishop, and white has castled short. Suppose that white had not done either of those, either had the light bishop or had castled long. Then the move c4 would not be good.
Why?
Here is the reason. By playing c4 black is doing several things. First he is gaining a little bit of space, but also he is controlling the d3 square. This is a hole in the white position, and since white does not have the light bishop, a piece of black's can one day use this square potentially. This is especially true because black still has the light bishop. It also extends the scope of the black dark bishop, something which may one day be useful if black goes for a queenside pawn attack. Finally, it denies the white knight on f3 of a square it would love to use, d4. d4 will now be forever home to a white pawn, and the knight on f3 is just that much worse.
Also, c4 is closing down the queenside, something that black is ok with because the white king is not on that side of the board. The action will be on the kingside, and black may continue with moves like Nd7 to f6, Be7, castles, and then the standard f6 break, opening the f-file and playing from there.
If you are interested in looking into this position further, I would encourage you to remove the knight on b1 and replace it with a bishop on e2. Then see what stockfish has to say about c4. I wouldn't be surprised if it doesn't like it anymore.
Be curious about the game, check out some different lines, and the meaning of the position will start to unfold. It helps to thoroughly understand one position, and thus understand many others that are similar.
there are other positions in the french where c4 loses for black. It depends on a lot of factors, but I think I touched on the big ones.
Also playing c4 in that particular position allows you to finish development (your knight via e7) without losing a pawn.
@chessfan124 another consideration is the location of your light square bishop. If he is inside the pawn chain, then c4 will block him in permanently. In this case, he is outside, and this makes c4 a better move.
Thanks everyone for your comments. Here's an example where I played c4 and it's really bad.
@chessfan124 in that position a3 was played so b3 was weak here a3 is not played so b3 is strong hope u understand:)
Thank you @Mahith1708 that does look like an important difference.
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