This is an attempt to inventory the different themes in the Polar Bear. It is partly based on what I encountered in my cc-games as attempt from white to prevent me from reaching the standard position. I only give the essentials of a position. This is the standard position:
Standard Polar Bear position.
Theme: preventing black from castling.
Option 1: Bc4
White can make it difficult for black to castle. Preconditions: white plays an early e3, while he omits c4.
The knight on f3 can coöperate with the bishop perfectly well. A knight on e6 would be killing. Black can intervene the beam from the bishop by playing e6, but Ng5 would force Qe7 which looks rather ugly. The prepatory move h6 leaves g6 vulnerable for Nh4. Probably black must keep the control over e6 by postponing d6 when Bc4 is in the air. That way black can play safely e6, castle, and regain a few tempi and get some space on the Queenside with c6, b5, a5 kicking the bishop. The blue square doesn't mean anything, it is a result of a bug in diagtransfer in combination with my screengrab utility.
Option 2: Qb3
Precondition: pawn c3.
The ideas are the same as with Bc4. Postpone d6, play e6 and castle. The Queen isn't well placed on b3 since it hampers white's queenside expansion.
Theme: inflict black with a double pawn on a6.
White plays Bxa6. Preconditions: e3 opens the diagonal, no c4 leaves it open, Nc3 forces c6, wich forces black to develop the knight to a6 instead of to c6, if g3 is played, the weak white squares around whites king can outweight blacks double pawn.
A possibility for black is to play a5 first and then Na6
Theme: pinning d6.
Precondition: white plays b3.
Black has prepared e5 in the usual way by Qc7, but he can't play it due to the pin of d6. Playing Re8 or d8 doesn't look nice since you want it to support f4 some day. Besides that there can come a moment you want to play Kh8. This leaves f7 weak for the white knight f3-g5-f7.
Theme: kingside attack.
Always be aware of a kingside attack with h4-h5. Especially when white hasn't castled or castles queenside.
It is remarkable how resilient blacks king position is often. But you must be vigilant! With a timely d5 and defensive assistance from Qc7 the problems are usually managable. A good timing of h6 and g5 can often keep the lines closed and the white knight out.
Theme: inflicting a double pawn on f6
White inflicts black with an awkward double pawn on f6 by playing Bxf6
Black must prevent that by playing e6 before g6 and Bg7.
Theme: exchanging black's blacksquared bishop.
White plays the bishop to e3, f4 or g5, followed by Queen d2 and Bh6 in order to trade the bishops with Bxg7
I haven't a conclusive solution for this theme yet. Black has to prevent the trade one way or another.
Theme: preventing e5.
If white plays Bf4 he hampers blacks pawn push to e5.
It is not clear if black can permit to play Qc7 to prepare e5 in this position. Especially if there is a white pawn on c4. Nh5 adds the bishop as support of e5, but Nxf4 doesn't diminish the amount of white defenders of e5 because of gxf4. Besides that, Bf4 can be a prepatory move for Qd2 and Bh6, intending to trade the bishops.
There are two other themes that play a role but that I haven't investigated thouroughly yet, I include them for the record.
Theme: white prepares an early e4.
White prepares the pawn push with Qe2, f3 and or Re1. It doesn't look very problematic for black.
Theme: black plays a stonewall.
Options: with Be7 or Bg7.
It is not clear to me if and when black is better off to play a stonewall.
This is a first rough inventory of the themes that play a role before you reach the standard position. None of these look very critical if black is vigilant. But a satisfactory solution must be found for every theme, since these typically are the bane of the patzer. Only when this part of the opening is mastered, I will look at the themes after the standard position.
10 hours ago