What is the purpose of 12.a3 in the QGD Lasker Defence?

Question

this is about queen gambit declined lasker defence :

[fen ""]
1. d4 d5 2. c4 e6 3. Nc3 Nf6 4. Bg5 Be7 5. e3 O-O 6. Nf3 h6 7. Bh4 Ne4 8. Bxe7 Qxe7 9. Rc1 Nxc3 10. Rxc3 c6 11. Qc2 Nd7 12. a3 

why or what is the purpose of 12.a3 ?

Answer 1

12. a3 in this position is a very concrete move, and as such, the best way to understand its purpose is to play out a handful of the main variations with and without the a3 inclusion. If it's difficult to do that on your own, keep an engine on your side while you're doing your exploratory analysis.

Short answer: a3 is not necessarily needed for white here, as there are alternatives such as Be2 (see Ivanchuk vs Carlsen 2012), but a3 helps keep a slight additional edge for white, as in a number of lines that can ensue white will have stopped any options of Qb4+, Qb5 (stopping white's castling) and Nf6-Nd5-Nb4 (specially in lines where black opts for b6.) To see why either of these options are relevant, let's go through some lines.

Longer answer (annotated, first go through the lines without a3):

 [title "Without and with a3 from move 12"]
 [fen "r1b2rk1/pp1nqpp1/2p1p2p/3p4/2PP4/2R1PN2/PPQ2PPP/4KB1R w K - 2 12"]

 1. Be2 (1. a3 {for the sake of comparison, let's follow through with both sides similarly to the Be2 line in order to highlight the underlying difference} dxc4 2. Bxc4 b6 3. Bd3 Nf6 (3...Bb7 {connecting the rooks and defending c6 directly} 4. Be4 Rfc8 5. Bxc6 {if c6-c5 could be played after Be4, black would be fine, but the knight on d7 leaves the queen and bishop disconnected, therefore, leaves c6 stuck on c6 and black cannot muster enough defense to hold on to it. So in this subvariation black still loses a clear pawn. We can conclude that after a3, black cannot really proceed with dxc4 and b6 ideas to finish development. However, dxc4 followed by e5 seems like a better alternative, but that's another discussion altogether.}) 4. Rxc6 Nd5 5. Qe2 Bb7 6. Rc1 {thanks to a3, white's a safe pawn up, has avoided simplifications for black, kept a healthy pawn structure and is ready to castle and maintaining control over the open file. Thus, a3, clearly refutes Nf6-Nd5 and leaves c6 hanging for black, so let's do better for black now.}) {Simply focusing on kingside development and to prepare short castles} dxc4 {a key forcing line that emphasizes the importance of control of b4 square} 2. Bxc4 b6 {in order for black to finish development, a solution has to be found for the locked out c8 bishop, therefore, either one opts for the e5 break early enough, or sets up the bishop on the long diagonal with b6, with either pawn breaks c5-e5 postponed for later.} 3. Bd3 {white's immediate reaction should be to prevent black's easy development, which here is simply to pressure the vulnerable c6 pawn, so Bd3 both uncovers the semi-open file and keeps the option of Be4 to add more pressure on c6.} Nf6 (3...Bb7 {for completeness let's consider the immediate Bb7 as well} 4. Be4 {usually throwing a Bh7+ first is good, but for the purpose of our discussions here it's not so important.} Rac8 5. Bxc6 Bxc6 6. Rxc6 Rxc6 7. Qxc6 Qb4+ {b4's not controlled by white!} 8. Qc3 Qb5 {black's well compensated for the pawn as white's unable to finish development and bring its last piece into play, black seems to be well in time for Nf6-Ne4-Nd6 followed by Rc8 claiming the open file, while white's left with no real initiative or even a clear path to finishing development.}) {but Bd3 is not even with tempo because black doesn't even have to immediately respond to the defense of c6 as Nf6 followed by Nd5 indirectly defends c6 as it comes with a double attack threat of Nb4 either forking the rook and queen, or the queen and bishop.} 4. Rxc6 Nd5 5. Qd2 {side-stepping the fork} Nb4 {b4's not controlled by white!} 6. Rc3 Nxd3+ 7. Qxd3 Bb7 8. O-O (8. Qe2 {as an alternative attempting not to prevent the kingside damage and holding on to the extra pawn} Rfc8 9. O-O Rxc3 10. bxc3 Rac8 11. Rc1 Qa3 12. Qd2 Bd5 {black's completely equalized as white's both weak pawns are under pressure and white has no initiative, Rc2 Be4 can even potentially lead to repetition.}) Bxf3 9. gxf3 Qg5+ 10. Kh1 Qd5 {black's winning back the temporarily sacrificed pawn and has a perfectly playable position without any weaknesses. This shows that Nf5-Nd5 is a very strong idea and if white wants to stop all these forcing variations which heavily simplify and equalize the game, then the said knight maneuver has to be stopped.}

Summary after our analysis: Very concretely, a3 puts a clear stop to dxc4 and b6 type of continuations for black (which would let them complete development comfortably), while maintaining the asymmetry of the position and avoiding simplifications which would favour black. Black on the other hand, still has a number of choices at their disposal after a3, one of which is dxc4 followed by e5, either way, black has to have planned for a well-timed pawn break with c5 or e5 if and when they decide to take on c4.

Answer 2

The main reason is to prepare a b4 push. This is especially useful if a Carlsbad structure is reached (after cxd5 exd5), when White will try to launch a queenside minority attack.

Another reason is to stop Black's queen from intruding on the square, as well as the b8-knight (e.g., ...Na6 followed by ...Nb4).