Fano $4$-folds with nef tangent bundle in positive characteristic

We employ the notation as in Proposition 2.7. Remark that $Y$ is ${\mathbb{P}}^{1}$. The contraction $g:X\to Z$ is a ${\mathbb{P}}^{1}$-bundle; moreover any fiber of $g$ is a section of $f$; this yields that $X$ is isomorphic to a product of ${\mathbb{P}}^{1}$ and $Z$.  

This follows from Proposition 2.10, Theorem 2.3 and the same argument as in [28, Proposition 5].  

See for instance [12].  

By [6, Theorem 5.1.3, Proposition 5.2.2], we see that

Then [6, Corollary 1.2.4] implies that ${\rm Br}(k)$ vanishes; thus our assertion follows from Proposition 2.13.  

By Theorem 2.3, we may find another smooth elementary contraction $f_{2}:X\to X_{2}$ besides $f_{1}$. Applying Theorem 2.2, Theorem 2.3 and [31], we see that $f_{2}:X\to X_{2}$ is a ${\mathbb{P}}^{1}$-bundle over $Q^{3}$ or a ${\mathbb{P}}^{2}$-bundle over ${\mathbb{P}}^{2}$. We claim that $f_{2}:X\to X_{2}$ is not a ${\mathbb{P}}^{1}$-bundle over $Q^{3}$. To prove this, assume that $f_{2}:X\to X_{2}$ is a ${\mathbb{P}}^{1}$-bundle over $Q^{3}$. Then by Corollary 2.14, $f_{1}$ and $f_{2}$ are given by the projectivizations of vector bundles. Let us consider the Chow ring of $X$. Since $f_{1}:X\to{\mathbb{P}}^{2}$ is a ${\mathbb{P}}^{2}$-bundle over ${\mathbb{P}}^{2}$, [9, Theorem 9.6] tells us that the rank of the $A^{3}(X)$ is three; however, since $f_{2}:X\to Q^{3}$ is a ${\mathbb{P}}^{1}$-bundle over $Q^{3}$, [9, Theorem 9.6] tells us that the rank of the $A^{3}(X)$ is two; this is a contradiction. As a consequence, $f_{2}:X\to X_{2}$ is a ${\mathbb{P}}^{2}$-bundle over ${\mathbb{P}}^{2}$. Applying [31], we conclude that $X$ is isomorphic to ${\mathbb{P}}^{2}\times{\mathbb{P}}^{2}$.  

Let $X$ be a smooth Fano $4$-fold with nef tangent bundle. Assume that $f_{1}:X\to{\mathbb{P}}^{3}$ is a ${\mathbb{P}}^{1}$-bundle. By Corollary 2.14, $f_{1}:X\to{\mathbb{P}}^{3}$ is given by the projectivization of a rank $2$ vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^{3}$, that is, $f_{1}:X={\mathbb{P}}({\mathcal{E}})\to{\mathbb{P}}^{3}$. We assume that ${\mathcal{E}}$ is normalized and consider its Chern classes $c_{1}({\mathcal{E}})$ and $c_{2}({\mathcal{E}})$ as integers $c_{1}$ and $c_{2}$ respectively. We denote by $H_{1}$ the ample generator of ${\rm Pic}({\mathbb{P}}^{3})$, by $F_{1}$ a fiber of $f_{1}$ and by $\xi$ the tautological divisor of ${\mathbb{P}}({\mathcal{E}})$. By the same argument as in [33, Theorem 2.1] and [11, Example 8.4.1] (see also [39]), we see that one of the following holds:

1. (i)

   ${\mathcal{E}}$ is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^{3}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{3}}$;

2. (ii)

   ${\mathcal{E}}$ is isomorphic to the null-correlation bundle ${\mathcal{N}}$;

3. (iii)

   ${\mathcal{E}}$ is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^{3}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{3}}(-1)$;

4. (iv)

   ${\mathcal{E}}$ is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^{3}}(-1)\oplus{\mathcal{O}}_{{\mathbb{P}}^{3}}(1)$;

5. (v)

   ${\mathcal{E}}$ is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^{3}}(-2)\oplus{\mathcal{O}}_{{\mathbb{P}}^{3}}(1)$;

6. (vi)

   ${\mathcal{E}}$ is a stable bundle with $(c_{1},c_{2})=(0,3)$;

7. (vii)

   ${\mathcal{E}}$ is a stable bundle with $(c_{1},c_{2})=(-1,4)$.

If ${\mathcal{E}}$ is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^{3}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{3}}(-1)$, ${\mathcal{O}}_{{\mathbb{P}}^{3}}(-1)\oplus{\mathcal{O}}_{{\mathbb{P}}^{3}}(1)$ or ${\mathcal{O}}_{{\mathbb{P}}^{3}}(-2)\oplus{\mathcal{O}}_{{\mathbb{P}}^{3}}(1)$, then $X={\mathbb{P}}({\mathcal{E}})$ admits a birational contraction, which contradicts to our assumption that the tangent bundle of $X$ is nef. To prove our assertion, it is enough to show that the cases (vi) and (vii) do not occur. In characteristic $0$, it was proved in [33, Theorem 2.1], but we do not know whether their argument also holds in positive characteristic or not.

From now on, we prove that the cases (vi) and (vii) do not occur. Since $X$ is a smooth Fano variety with nef tangent bundle, Theorem 2.3 tells us that there exists another smooth elementary contraction $f_{2}:X\to X_{2}$. For any ample divisor $H_{2}\in{\rm Pic}(X_{2})$, there exist integers $a,b$ such that

Let us first assume that ${\mathcal{E}}$ is a rank $2$ stable bundle on ${\mathbb{P}}^{3}$ with $(c_{1},c_{2})=(0,3)$. We claim that $a$ and $b$ are not equal to $0$. Taking the intersection number of $a\xi+bf_{1}^{\ast}H_{1}$ and $F_{1}$, we have

Moreover, if $b=0$, then we have ${\mathcal{O}}_{X}(f_{2}^{\ast}H_{2})\cong{\mathcal{O}}_{X}(a\xi)\in{\rm Pic}(X)$; in particular $\xi$ is nef; thus ${\mathcal{E}}$ is nef and $c_{1}=0$. By Corollary 2.5 (iii), ${\mathcal{E}}$ is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}^{3}}^{\oplus 2}$; this is a contradiction. As a consequence, we see that $b\neq 0$.

Since we have $\xi^{2}+c_{2}f_{1}^{\ast}H_{1}^{2}=0$, $f_{1}^{\ast}H_{1}^{4}=0$ and $\xi\cdot f_{1}^{\ast}H_{1}^{3}=1$, we also have

By using these equations, we have

Since $a,b\neq 0$, we have

However this is a contradiction.

Secondly assume that ${\mathcal{E}}$ is a rank $2$ stable bundle on ${\mathbb{P}}^{3}$ with $(c_{1},c_{2})=(-1,4)$. By the same argument as in the case where ${\mathcal{E}}$ is stable and $(c_{1},c_{2})=(0,3)$, we have

Thus we have an equation

Since $a$ is positive, we have

Then it follows from the rational root theorem that $\dfrac{\,b\,}{a}=\dfrac{\,1\,}{2}$; this implies that $2\xi+f_{1}^{\ast}H_{1}$ is nef. Since $X={\mathbb{P}}({\mathcal{E}})$ is a Fano variety, we have

This is a contradiction. As a consequence, our assertion holds.  

Let $X$ be a smooth Fano $4$-fold with nef tangent bundle. Assume that $f_{1}:X\to X_{1}:=Q^{3}$ is a ${\mathbb{P}}^{1}$-bundle. To prove Proposition 3.4, assume the contrary; then we have another ${\mathbb{P}}^{1}$-bundle structure $f_{2}:X\to X_{2}:=Q^{3}$ besides $f_{1}$. By Corollary 2.14, $f_{i}:X\to X_{i}$ ($i=1,2$) is given by the projectivization of a rank $2$ vector bundle ${\mathcal{E}}_{i}$ on $Q^{3}$. Denoting by $\xi_{i}$ the divisor corresponding to the tautological line bundle ${\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}}_{i})}(1)$ and by $H_{i}$ the ample generator of ${\rm Pic}(X_{i})$, we have

Remark that the Chow groups $A^{1}(Q^{3})$ and $A^{2}(Q^{3})$ are isomorphic to ${\mathbb{Z}}$; then we consider the Chern classes $c_{1}({\mathcal{E}}_{1}),c_{1}({\mathcal{E}}_{2}),c_{2}({\mathcal{E}}_{1})$ and $c_{2}({\mathcal{E}}_{2})$ as integers $c_{1},c_{1}^{\prime},c_{2}$ and $c_{2}^{\prime}$ respectively:

We may assume that ${\mathcal{E}}_{i}$’s are normalized, that is, $c_{1},c_{1}^{\prime}\in\{-1,0\}$. We denote by $F_{i}$ a fiber of $f_{i}$.

By Proposition 2.7, $f_{2}$ is nothing but the elementary contraction which contracts minimal birational sections of $f_{1}$ over lines on $Q^{3}$; this implies that

Then we obtain

Comparing the parity of both sides of this equation, we see that $c_{1}=-1$; thus we obtain

Since we have $\xi_{1}^{2}+\xi_{1}\cdot f_{1}^{\ast}H_{1}+\dfrac{1}{2}c_{2}f_{1}^{\ast}H_{1}^{2}=0$, $f_{1}^{\ast}H_{1}^{4}=0$ and $\xi_{1}\cdot f_{1}^{\ast}H_{1}^{3}=2$, we also have

For the ample generator $H_{2}$ of ${\rm Pic}(X_{2})$, there exist integers $a,b$ such that ${\mathcal{O}}_{X}(f_{2}^{\ast}H_{2})\cong{\mathcal{O}}_{X}(a\xi_{1}+bf_{1}^{\ast}H_{1})\in{\rm Pic}(X)$. By this definition, $(a,b)\neq(0,0)$. Moreover we have

Hence, we obtain $a=b\neq 0$; then we obtain

Hence we obtain $c_{2}=1$. Let us take integers $\alpha,\beta,\gamma,\delta$ as follows:

Remark that $|\alpha\delta-\beta\gamma|=1$, because $\{\xi_{i},f_{i}^{\ast}H_{i}\}$ is a ${\mathbb{Z}}$-basis of ${\rm Pic}(X)$ for $i=1,2$. Since we have

and

we obtain

Then the equality $|\alpha\delta-\beta\gamma|=1$ implies that $f_{2}^{\ast}H_{2}\cdot F_{1}=1$. Computing $-K_{X}\cdot F_{1}$, we see that $c_{1}^{\prime}=-1$ and $\xi_{2}\cdot F_{1}=-1$. Thus the equation (2) can be written as follows:

Now we have

This is a contradiction.