 Solution

One useful version is the effective global Torelli theorem for K3 surfaces. Let $X,Y$ be complex K3 surfaces and letpreserve the integral cup-product pairing and the Hodge decomposition. Suppose that $\varphi$ carries some Kähler class of $Y$ to a Kähler class of $X$. ThenConversely every isomorphism has these properties. In particular, for projective K3 surfaces an integral Hodge isometry carrying an ample polarization to an ample polarization is induced by a unique isomorphism of the polarized surfaces. This is the effective global Torelli statement.

The Kähler-cone condition is necessary if the prescribed isometry itself is to be induced by an isomorphism. If $\delta$ is the class of an effective smooth rational $(-2)$-curve, the reflectionis integral, preserves the pairing, and preserves the Hodge decomposition because $\delta$ has type $(1,1)$. It sends $\delta$ to $-\delta$. An automorphism preserves effectivity, whereas $-\delta$ is not effective: it has negative intersection with an ample class. Thus this Hodge isometry is not an automorphism pullback. Requiring an isometry to respect the Kähler chamber removes precisely this kind of obstruction.