White has a clear advantage, even though it's opposite coloured bishops:
He has a extra pawn
second, the queenside pawns existence, if they were not there, you could sac your h-pawn and get to a theoretical drawn ending, with the idea of sacrificing your bishop for the 2 pawns. Thats not a good idea here, since white then win with his queenside pawns.
third, 2 connected passed pawns, while you only got 1.
With 38: Kg3!!, it seems hard to defend because the king becomes extremely active. But after 38: Bf4?! h4! (Bf7 also works, according to fritz, however this is best)
From here, there are 2 plans for white:
1: Trying to take the h-pawn
2: Activating the king, trying to get his own queen, and ignoring the h-pawn
Plan 1: 39: Kg2!? Bh5! 40: Bb8 a6 41: f4 Kg6! (now 42: Kh3?! is answered by Bf3! 43: e5 Kh5 with an easy draw) 42: Bd6 Bd1 43: f5+ (only way to continue) 43... Kf6 44: Bh2! (Kh3?! Bc2 with a draw) 44: ... Bc2 45: Kf3 Kg5! (only drawing move) 46: Bc7 Bd1+ (the threat was Be7+) 47: Ke3 Kg4 48: Kd4 h3! (not allowing 49: Kd4, since h2 then wins for black! The resulting position looks defendable.
Plan 2 doesn't really work, as black can at a point play h3, transposing to a draw.
More interesting is 38: Kg3! The position looks like a practical disaster for black, and holding it theoretically can be hard. White can penetrate with Kf4 or Kh4, and should easily win: 38: ... Ke5 (only move not allowing Kf4) 39: Bg7+ Ke6 40: Kh4! The pawns and the bishop takes alot of squares from the black king. 40: ... Kf7 41: Bd4 Ke7 42: Kg5 Bf7 43: Be5! b6!? Trying to generate counterplay) 44: Bd6+ Kd7 45: f4 bxc5 46: Bxc5 a6 47: f5 with an easy win.
From a theoretical point of view, after 38: Bf4?! white has nothing, but practically he should try to win. It's hard to defend as black
A few points. 1: Viktor is absolutely correct that white has a pretty serious advantage here. Ignore the queenside pawns for a moment, they don't matter. White's got a pair of connected passers, which (as he showed) he can ram down your throat and even with your bishop there's precious little you can do about that.2: Opposite colored bishops do not de-facto translate into a drawn endgame. What OCBs do is make promoting a SINGLE pawn very difficult. The player with the advantage cannot threaten a bishop trade to drive the enemy piece away from a square, and thus is vulnerable.3: I'm pretty sure this is a forced win for white... but for my rating I'm a pretty weak endgame player. I do see one strategy that might draw, but that's the best I can do. However a better player than I could probably *glance* at the board for 2-3 seconds and instantly tell you if my idea works or not :-/. Always humbling when that happens.As for that idea, It's to push the h pawn as soon as possible. Ideally you want him on h3, forcing white to defend the h2 square or else you promote. With a piece tied up on your h pawn, you should be able to impede white's passers and work your way into a drawn position.
Given the starting position above, I think draw is achievable for black. The general idea is to place the Black King at e5 as soon as possible. Then try to overpower and eventually eliminate White's central pawns while pushing the h pawn forward to create distraction.
Note that white's bishop cannot protect his e4 pawn, whereas black can attack it with both his king and bishop. Hence, as long as black keeps pressurizing the e4 pawn, while pushing the h pawn for distraction, draw is achievable.
Just realized there was an obvious blunder in the line I posted. Re-try: