I'm reading John Nunn's book Grandmaster Chess Move by Move, and it stresses the importance of the h4 square in the following endgame.
8/2p2p2/5k1p/2B1p1p1/4P3/1P3P2/2n1K1PP/8 w - - 0 1 1. Kd2? Nd4 2. Bxd4 exd4 3. b4 Ke5 4. Kd3 h5! 5. g3 h4
and [Black] is just in time to prevent h4 by White. Now White cannot make progress and the game is drawn.
Using computer analysis, I see that the move 6. Kc4 loses in the resulting position, due to 6... hxg3 7. hxg3 g4! So white's best attempt is not Kc4 but rather playing f4 at the right time.
Why is it advantageous for each side to occupy h4? I have some thoughts on the matter but would appreciate a more complete answer.
If white occupies h4, he can exchange hxg5 hxg5 at the right time and liquidate the kingside, so that black can't queen with an h-pawn. This is especially important since white's plan is to push f4, capture the d4 pawn and then attack on the queenside. Better not to leave behind a black h-pawn that can queen.
If black occupies h4, then he has an advanced h-pawn that can potentially queen, especially if white tries to break through with f4. (In the case of f4, black can counterattack on the kingside.)
Compare the above (drawn) position with the position below which is won for white. The moves are given by Stockfish. I think this illustrates why exchanging h-pawns is advantageous for white, so that he can attack on the queenside more freely.
8/2p2p2/7p/4k1p1/1P1pP2P/3K1PP1/8/8 w - - 0 1 1. hxg5 hxg5 2. b5 f6 3. f4+ Ke6 4. Kxd4 Kd6 5. e5+ fxe5+ 6. fxe5+ Ke6 7. Kc5 Kxe5 8. Kc6 Ke4 9. Kxc7 Kf3 10. b6 Kxg3 11. b7 Kf2 12. b8=Q g4 13. Kc6 Kg2 14. Kd5 g3 15. Qf8 Kh2 16. Ke4 g2 17. Qf4+ Kg1
What are your thoughts on the endgame? Why do you think the h4 square is important for both sides? Am I leaving something out in my analysis?