Is the projective space over matrices of rank bigger than $k$ compact?

Question

Let $V$ be a real $d$-dimensional vector space ($d>2$). Let $2 \le k \le d-1$ be fixed. Define $H_{>k}=\{ A \in \text{End}(V) \mid \operatorname{rank}(A) > k \}$. $H_{>k}$ is an open submanifold of $ \text{End}(V)$.

Now consider the projective space $\mathcal{P}\big(H_{>k} \big)$, obtained from $H_{>k}$ by dividing by the equivalence relation $A \sim \lambda A$ ($\lambda \in \mathbb{R}$ is non-zero).

Is $\mathcal{P}\big(H_{>k} \big)$ compact?

The standard proof showing the projective space $\mathbb{R}P^n$ is compact goes by restricting the quotient map from the whole vector space to the sphere; that is if we denote by $\pi:\mathbb R^{n+1}\to\mathbb RP^n$ the quotient map, then $RP^n=\pi(\mathbb{S}^n)$ is the image of a compact set, hence compact.

In our case the ambient vector space is $\text{End}(V) \simeq\mathbb{R}^{d^2}$. However, $H_{>k} \cap \mathbb{S}^{d^2-1}$ is not closed in $\text{End}(V) $, since the rank can fall in the limit. Thus, even though $\mathcal{P}\big(H_{>k} \big)=\pi\big(H_{>k} \cap \mathbb{S}^{d^2-1}\big)$ we cannot conclude from this it is compact.

Of course, this does not show $\mathcal{P}\big(H_{>k} \big)$ is not compact...

Answer 1

$H_{>k}$ is the set of matrices such that at least one of the submatrices of order $k+1$ has non-zero determinant. So $H_{>k}$ is actually an open cone in $\Bbb R^{n\times n}\setminus\{0\}$. Therefore, its projection is open in $\Bbb {RP}^{n^2-1}$ and, by connectedness, not closed. Nor, of course, compact.

Answer 2

The complement of $H_{>k}$ is closed, so you are essentially removed a non-empty closed subset of a projective space. The projection is an open mapping, meaning that $H_{>k}$ is non compact.

Answer 3

No. Indeed, the inclusion $H_{>k}\to \operatorname{End}(V)\setminus\{0\}$ induces an embedding $\mathcal{P}(H_{>k})\to \mathbb{P}(\operatorname{End}(V))$. The image of this embedding is not closed (it is a dense proper subset), so $\mathcal{P}(H_{>k})$ is not compact.