Global Torelli theorem for K3 surfaces ($\varphi =f^{\ast }:H^2(Y,\mathbb{Z})\to H^2(X,\mathbb{Z})$)

An integral Hodge isometry between complex K3 surfaces which carries some Kähler class to a Kähler class is the pullback of a unique isomorphism of the surfaces. A Hodge isometry without that cone condition need not itself be geometric: reflection in an effective $(-2)$-curve reverses its class and cannot be induced by an isomorphism.