The weak Lefschetz principle is false for ample cones. (Q700536)

The weak Lefschetz theorem asserts that \(i^{*}:H^{k}(X,\mathbb Z)\rightarrow H^{k}(D,\mathbb Z)\) is an isomorphism compatible with the Hodge decomposition for \(k\leq n-1\). Here \(X\) is an \((n+1)\)-dimensional smooth complex variety, \(D\) a smooth ample divisor and \(i\) an inclusion. One of the consequences of this theorem is that \(i^{*}: \text{Pic}(X)\rightarrow \text{Pic}(D)\) is an isomorphism for \(n\geq 3\). The authors consider the following question: Let \(\mathcal L \in \text{Pic}(X)\) be such that \(i^{*}(\mathcal L)\) is ample on \(D\). Is it true that \(\mathcal L\) must be ample on \(X\)? They built two counterexamples showing that this is not true in general and provide an example of the situation for which the answer is positive.