The Weak Lefschetz Principle is False for Ample Cones

Let X be an (n + 1)-dimensional smooth complex projective variety and let D be a smooth ample divisor of X with inclusion map i : D → X. The wellknown Weak Lefschetz Theorem (see [GrHa]) asserts that the restriction map i∗ : H(X;Z) → H(D;Z) is an isomorphism for k ≤ n−1 and is compatible with the Hodge decomposition. For n ≥ 3 one deduces from these results that i∗ : Pic(X) → Pic(D) is also an isomorphism. Grothendieck has shown that this statement is true over any algebraically closed field [Hart]. While an ample line bundle on X always restricts to an ample line bundle on D, it is not at all clear whether Amp(D) ⊂ i∗Amp(X), i.e., whether the Weak Lefschetz Principle holds for the ample cone. In this note we provide two examples showing that the Weak Lefschetz Principle for the ample cone fails in general. One is obtained by blowing up (§2) and the other is a product with a P factor (§3). We also provide some partial positive results (§4). However, a complete picture of how the ample cone behaves under the Weak Lefschetz isomorphism remains elusive.