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,
.
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
.
and
and it is a counterexample.
. One can use the fact that even dimensional quadric has
as middle cohomologies.
. The same (
) is true for all quadrics and all Grassmannians.
I claim that
To obtain the first equality we need to show that
The second equality follows from the fact that
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
.
Sorry, I said it unclear in the main post. I want an example with a singular but still irreducible
.