## Looking for a better counterexample

This post is doubling my old question asked on Mathoverflow.

Let be a smooth projective complex variety of dimension , and let be an ample irreducible divisor. Recall that a cohomology class is called -primitive, if . Here, is the first Chern class of the line bundle associated to . We denote by the subset of primitive elements.

Let denote by the complement to , and by the open embedding. I claim that if is smooth then the composition is an injection. I skip a proof; this claim is an exercise on the Gysin exact sequence and the Lefschetz theorems. Perhaps, it is more interesting to find counterexamples with a singular . In my old question, I came up with something strange, and I hope that now somebody can give me a better example.

My construction goes as follows. Let be a 3-dimensional vector space. Consider embedded in by Veronese. Let be the Grassmann variety parameterizing 3-dimensional planes in , so . Let be the subset of planes which intersect embedded non-transversally. One can show that is irreducible. Moreover, is ample because any effective divisor in is ample.

I claim that the complement is the configuration space of four non-ordered points in in general position. Skipping details, it follows from the fact that any four such points in can be given as an intersection of two quadrics. Nevertheless, the group acts on , and this action is transitive and with finite stabilizers as it follows from the previous description. Thus, In particular, . However, .

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I shall write the simplest example I know but you can generalize it to all even dimensional smooth quadric (except surface) or to any Grassmannian except . Let be a smooth quadric in or the same . We take as the tangent hypeplane to a point .
I claim that and and it is a counterexample.
To obtain the first equality we need to show that . One can use the fact that even dimensional quadric has as middle cohomologies.
The second equality follows from the fact that . The same ( ) is true for all quadrics and all Grassmannians.
Can’t we simply take to be a union of several divisors on a surface whose classes span all of ? For instance, take with . This divisor is ample and the class is primitive yet it surely dies in .