Embeddings of lens spaces in complex projective plane $\Bbb CP^2$

Question

Is it possible to embed a lens space $L(p,q)$ (smoothly or topologically) into the complex projective plane $\Bbb CP^2$? According to this note https://web.ma.utexas.edu/users/aissa/slides/candidacy.pdf, it is impossible to embed $L(p,q)$ with $p>1$ in $S^4$. So I am curious about the same problem with $S^4$ replaced by $\Bbb CP^2$, or blow-ups $\Bbb CP^2\sharp n\overline{\Bbb CP^2}$ of $\Bbb CP^2$. Any comments or references will be thankful!

Answer 1

Some lens spaces embed in $\mathbb{C} P^2$ and some don't and the question of exactly which lens spaces embed in $\mathbb{C} P^2$ is open for both smooth and topological embeddings. However, some restrictions are known.

The easiest obstruction to embedding in $\mathbb{C} P^2$ is the following. If a lens space $L$ embeds smoothly (resp. topologically) into $\mathbb{C} P^2$, then $L$ bounds a smooth (resp. topological) rational homology ball. This is because the image of $L$ separates $\mathbb{C} P^2$ into two components. Applying the Mayer-Vietoris sequence with rational coefficients shows that one of these components is a rational homology ball.

I don't have much to add about topological embeddings other than to note that the above obstruction implies that the lens space $L(p,q)$ can only embed topologically in $\mathbb{C} P^2$ only if $p$ is a square.

If one is interested in smooth embeddings into $\mathbb{C} P^2$, then the work of Lisca which classifies which lens spaces bound smooth rational homology balls. This gives a list of lens spaces which can potentially be embedded in $\mathbb{C} P^2$. However, not every lens space on Lisca's list embeds in $\mathbb{C} P^2$. For example, Brendan Owens has an obstruction based on Donaldson's diagonalization theorem that can obstruct these embeddings. See for example Example 3.4 of this paper. His paper also provides some examples of lens spaces which do embed smoothly into $\mathbb{C} P^2$.