degrees of complex projective spaces and quadrics

Question

A well-known result of Kobayashi and Ochiai says that an $n$-dimensional Fano maniofold $M$ is biholomorphic to $\mathbb{C}P^n$ or complex quadrics if its index is $n+1$ or $n$ respectively. In these two cases, the degrees are $(n+1)^n$ and $2\cdot n^n$. My question is, if the degree of an $n$-dimensional Fano manifold is exactly $(n+1)^n$ (resp. $2\cdot n^n$), can we conclude that the manifold is $\mathbb{C}P^n$ or the quadrics? I know there is also a famous inequality $\text{degree}\leq(n+1)^n\cdot\frac{n+1}{\text{index}}$ and the equality holds iff it is $\mathbb{C}P^n$. But it seems that this inequality cannot be applied to give a negative answer to my question.

Answer 1

Apparently, there used to be a related conjecture

No Longer a Conjecture Let $X$ be a (smooth projective) Fano manifold of dimension $n$. Then $c_1(X)^n \leq (n+1)^n$ with equality only if $X\simeq \mathbb P^n$.

Apparently Batyrev (Boundedness of the degree of multidimensional toric Fano varieties. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1982, no. 1, 22–27, 76–77) showed that this inequality fails (which actually does not mean that what the OP asked fails). The counterexample has Picard number $>1$, and as far as I can tell it is possible that this is true with Picard number $=1$, but I don't think it is known.

The following inequality is known:

Theorem (Nadel, Campana, Kollár-Miyaoka-Mori): Let $X$ be a (smooth projective) Fano manifold of dimension $n$ over an algebraically closed field of characteristic $0$. Assume that $\mathrm{Pic}\, X\simeq \mathbb Z$. Then $$ c_1(X)^n \leq \big( n(n+1) \big)^n. $$

This was proved by the list of authors independently around the same time. It is included as V.2.2 in Rational Curves on Algebraic Varieties by J. Kollár.

I don't know if anyone has thought about the actual question here without the inequality, but it seems natural from the point of other results. I don't have a proof either way, but I have some ideas:

The following generalization of the Kobayashi-Ochiai theorem of Cho--Miyaoka--Shepherd-Barron (3 people) may be relevant:

Theorem Let $X$ be a (smooth projective) Fano manifold of dimension $n$ and assume that $c_1(X)\cdot C\geq (n+1)$ for any (proper) curve $C\subset X$. Then $X\simeq \mathbb P^n$.

You can find the original article in this book, but it might be easier to read this proof by Kebekus.

Another possibly relevant statement is the following

Proposition Let $X$ be a smooth projective variety of dimension $n$ and $\mathscr L$ an ample line bundle on $X$. Assume that for some $d\in\mathbb N$ there is a point $x\in X$ such that any general point $y\in X$ can be connected to $x$ by an irreducible rational curve $C_{y}$ with $c_1(\mathscr L)\cdot C_y \leq d$. Then $c_1(\mathscr L)^n\leq d^n$.

This is much easier than the above. It is proved for example in V.2.9 Kollár's book.

As promised, this does not give you a proof, but suggests that there might be some interesting statement. Namely, by Bend & Break (see II.5 of Kollár's book) for every $x\in X$ there exists an irreducible rational curve $C\subset X$ such that $x\in C$ and $c_1(X)\cdot C \leq (n+1)$. In other words, the Cho--Miyaoka--Shepherd-Barron theorem says that if $(n+1)$ is indeed the smallest value possible to satisfy this inequality, then $X\simeq \mathbb P^n$.

On the other hand, if there is a point for which the minimal degree rational curves going through that point cover a dense part of $X$ and this minimal degree is $d< (n+1)$, then by the above Proposition it follows that $c_1(X)^n\leq d^n < (n+1)^n$, so $X\not\simeq \mathbb P^n$. Of course, it is possible that the minimal degree rational curves do not dominate $X$, but Mori's proof of the Hartshorne conjecture (see V.3.2 of Kollár's book) suggests that it might not happen for such high minimal degree. At the same time, the counterexample of Batyrev and the works mentioned by Ruadhai in the comments suggest that one would probably need to assume more such as Picard number one.

Anyway, this is getting pretty long for an answer without an answer and so far I have only addressed the projective space case and not the quadric. For that the situation is similar: there is a paper by Miyaoka proving the equivalent of the above Theorem for quadrics and the pseudo-argument about minimal degree rational curves applies the same way.

There are also plenty of related results. Some are referenced in the Cho--Miyaoka--Shepherd-Barron paper as well as in Kebekus's. The background to all of this is included in Kollár's book. More recent related papers include this one by Andreatta-Wiśniewski, this one by Araujo and this one by Araujo-Druel-Kovács.