varieties with no new cohomology in hyperplane sections

Question

Let $X$ be a smooth closed subvariety in a complex projective space and $Z\subset X$ be a generic hyperplane section. Lefschetz hyperplane section Theorem says that the restriction map $H^i(X)\to H^i(Z)$ is an isomorphism for $i<dim(Z)$ and injective for $i=dim(Z)$. I am interested in examples when it's also an isomorphism $i=dim(Z)$. This is equivalent to saying that Fourier-Deligne transform of the intersection cohomology sheaf of the cone over $X$ has less than full support in the dual space (then this support has to be the cone over the projectively dual variety $X^*$). This is so when $X$ is an even dimensional quadric; are there other examples?

Answer 1

If $\dim X=2$, this is the case if and only if $X$ is a ruled surface (the quadric included), $X=v_2(\mathbb P^2)$, or $X$ is an isomorphic projection of $v_2(\mathbb P^2)$ in $\mathbb P^4$.

This is also the case if $\dim X>2$ and the dual variety $X^*\subset(\mathbb P^N)^*$ is not a hypersurface (``new cohomology'' is spanned by vanishing cycles, and if $X^*$ is not a hypersurface, then there are no vanishing cycles since Lefschetz pencils have no degenerate fibers).

If $\dim X>1$ is odd, the converse is true. Indeed, in this case vanishing sycles have self-intersection $\pm 2$, hence they are never zero if they exist.

In small dimensions, a classification of varieties with small duals is known; see two papers by L.Ein titled ``On varieties with small dual varieties''.

Answer 2

The simplest example, of course, is a projective space.

Another one, is the double Veronese embedding of an even-dimensional projective space.

Yet another one is $Gr(2,5)$ (here you can iterate up to three hyperplane sections).

There are also other examples.