Big eyes can be small in the corner
Even if White goes first, White falls two moves short: Black can make an eye inside White's big eyespace due to the special properties of the corner, giving atari. This way, instead of five liberties, White has only three.
Preventing the eye with here makes matters even worse. If we take the - exchange into account, White has two liberties.
Black can exchange for , because otherwise Black would make an eye in the corner. After the exchange, we see that White has three liberties (that is, 5 - 2) against Black's five. White is dead even if White has sente.
Important to notice that the white eye still has its quality as a big eye if there are shared liberties.