Let $X$ be a smooth $n$-dimensional complex projective variety and $D \subset X$ a smooth (effective) divisor. Consider the following **properties:**

- $D$ is
**ample.** For any $k$-dimensional subvariety $Z \subset X$ the intersection $D^{n-k} \cdot Z$ is strictly positive.*(Positivity)*For any $k$, $1 \le k \le n$ the map $$\cup [D]^k \colon H^{n-k}(X, \mathbb{C}) \to H^{n+k}(X, \mathbb{C})$$ is an isomorphism.*(Hard Lefschetz)*If $j \colon D \hookrightarrow X$ is the tautological embedding, the induced maps $$j_* \colon H_k(D, \mathbb{Z}) \to H_k(X, \mathbb{Z})$$ are isomorphisms for $k<n-1$ and surjective for $k=n-1$.*(Weak Lefschetz / Lefschetz Hyperplane)*The induced maps $$j_* \colon \pi_k(D) \to \pi_k(X)$$ are isomorphisms for $k<n-1$ and surjective for $k=n-1$.*(Zariski-Lefschetz / Homotopic Lefschetz Hyperplane)*

Of course, the first property implies all the rest (moreover, I guess this is the reason why ample divisors play such important role in algebraic geometry). It is also well-known, that 2) imples 1) *(Kleiman criterion)*.So my **questions** are the following:

- Does 3) implies ampleness?
- Does 4) implies ampleness?
- It seems that Hard Lefschetz implies Weak one, however it is not clear to me, if it is equivalent. I am also not really sure, if 3) holds when one replaces cohomology with coefficients in $\mathbb{C}$ by integral cohomology.
- It is absolutely non-clear, if Hard Lefschetz implies 5. It would be very interesting to know if there are examples, when 4) holds, but 5) does not.

(**Few remarks**: It would be also interesting to know the answers to this questions for arbitrary ground-field, as far as they make sense.

For me, the most intriguing question from the list is "Does weak Lefschetz implies ampleness?". Certainly, it needs some additional discussion. First of all, there are examples due to B. Totaro when two smooth divisors from the same cohomology class have different topology, which can not happen with smooth divisors. Secondly, I know few boring counter-examples such as $C \times \{0\} \subset C \times \mathbb{P}^1$, where $C$ is any non-rational curve. May be one needs to ask $n \ge 3$ and/or simply-connectedness of $X$ plus some extra conditions to escape Totaro's-type issues.)

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