A theorem of ample subsheaf of tangent ample
In my previous post, I mentionned about a generalization of Hartshorne's conjecture in [Jie17] :
Theorem 1: Let $X$ be a complex projective manifold of dimension $n$. Suppose
that $T_X$ contains an ample subsheaf $\mathcal{F}$ of positive rank $r$, then $(X, \mathcal{F})$ is isomorphic to $(\mathbb{P}^n, T_{\mathbb{P}^n})$ or $(\mathbb{P}^n, O_{\mathbb{P}^1}^{\oplus r} )$.
This theorem only needs $T_X$ containing an ample subsheaf, which is far stronger than theorem 8 in [Mor79]. To prove Theorem 1, Jie represented the concept of foliation and then combined with Theorem 1.1 in [Arau06] to derive to two key theorems, those are 2.2.9 and 2.3.8 in [Jie17]. For detail of proofs and definitions, please c.f. [Jie17].
Now we will come to key theorems for our proofs of Theorem 1:
Theorem 2 (theorem 2.2.9): Let $X$ be a projective manifold. Assume that $T_X$ contains an ample subsheaf $\mathcal{F}$ of positive rank $r < dim(X)$. Then its saturation $\overline{\mathcal{F}}$ defines an algebraically integrable foliation on $X$, and the closure
of the $\overline{\mathcal{F}}$-leaf passing through a general point is isomorphic to $\mathbb{P}^r$ ..
Theorem 3 (theorem 2.3.8): Let $X$ be a $n$-dimensional projective manifold. If $\mathcal{F} \subset T_X$ is an ample subsheaf of positive $rank r < n$, then there is a common point in the closure of general leaves of $\mathcal{F}$.
Proof of theorem 1. Theorem 1.1 in [Ara06] implies that there exists an open subset $X^0 \subset X$ and a variety $T^0$ such that $X^0 \rightarrow T^0$ is a $\mathbb{P}^{r+1}$-bundle and $d+1 > r$. Assume $r < n$. By theorem 2 and theorem 3, we have $\overline{\mathcal{F}}$ define a algebraic integrable foliation over $X$ such that every closure of $\overline{\mathcal{F}}$-leaves have a common point. Assume contrary that $dim T^0 \geq 1$. Take any fibre $F$ of $\varphi: X^0 \rightarrow T^0$, an arguement in the proof of theorem 2 shows that $(F, \mathcal{F}_{|F})$ is isomorphic to $(\mathbb{P}^{d+1},T_{\mathbb{P}^{d+1}})$ and hence define an algebraically integrable foliation on $X^0$ with $\mathcal{F}_{|F}$-leaves which is also a $\mathcal{F}_{|X^0}$-leaves (by ex 2.2.4 and definition in [jie17]). For a different fibre $F'$, by the same statement, $\mathcal{F}_{|F}$ defines an algebraically integrable foliation on $X^0$ with $\mathcal{F}_{|X^0}$-leaves and combines with previous argument, we will get two projective subvariety on two fibre $F$ and $F'$ have a common point by theorem 3 (applies for two $\mathcal{F}_{|X_0}$-leaves which lie in $F$ and $F'$ respectedly), which is impossible because any of two fibre can't have a non-zero intersect. Hence, $dim T^0 = 0$ and by lemma 1, $X \cong \mathbb{P}^n$.
Conferences:
[Jie17] Jie Liu. Géométrie des variétés de Fano: Sous - Faisceaux du fibré tangent et diviseur fondomental. Thèse De Doctorat, Université Côte d'Azur.
Theorem 1: Let $X$ be a complex projective manifold of dimension $n$. Suppose
that $T_X$ contains an ample subsheaf $\mathcal{F}$ of positive rank $r$, then $(X, \mathcal{F})$ is isomorphic to $(\mathbb{P}^n, T_{\mathbb{P}^n})$ or $(\mathbb{P}^n, O_{\mathbb{P}^1}^{\oplus r} )$.
This theorem only needs $T_X$ containing an ample subsheaf, which is far stronger than theorem 8 in [Mor79]. To prove Theorem 1, Jie represented the concept of foliation and then combined with Theorem 1.1 in [Arau06] to derive to two key theorems, those are 2.2.9 and 2.3.8 in [Jie17]. For detail of proofs and definitions, please c.f. [Jie17].
Now we will come to key theorems for our proofs of Theorem 1:
Theorem 2 (theorem 2.2.9): Let $X$ be a projective manifold. Assume that $T_X$ contains an ample subsheaf $\mathcal{F}$ of positive rank $r < dim(X)$. Then its saturation $\overline{\mathcal{F}}$ defines an algebraically integrable foliation on $X$, and the closure
of the $\overline{\mathcal{F}}$-leaf passing through a general point is isomorphic to $\mathbb{P}^r$ ..
Theorem 3 (theorem 2.3.8): Let $X$ be a $n$-dimensional projective manifold. If $\mathcal{F} \subset T_X$ is an ample subsheaf of positive $rank r < n$, then there is a common point in the closure of general leaves of $\mathcal{F}$.
Proof of theorem 1. Theorem 1.1 in [Ara06] implies that there exists an open subset $X^0 \subset X$ and a variety $T^0$ such that $X^0 \rightarrow T^0$ is a $\mathbb{P}^{r+1}$-bundle and $d+1 > r$. Assume $r < n$. By theorem 2 and theorem 3, we have $\overline{\mathcal{F}}$ define a algebraic integrable foliation over $X$ such that every closure of $\overline{\mathcal{F}}$-leaves have a common point. Assume contrary that $dim T^0 \geq 1$. Take any fibre $F$ of $\varphi: X^0 \rightarrow T^0$, an arguement in the proof of theorem 2 shows that $(F, \mathcal{F}_{|F})$ is isomorphic to $(\mathbb{P}^{d+1},T_{\mathbb{P}^{d+1}})$ and hence define an algebraically integrable foliation on $X^0$ with $\mathcal{F}_{|F}$-leaves which is also a $\mathcal{F}_{|X^0}$-leaves (by ex 2.2.4 and definition in [jie17]). For a different fibre $F'$, by the same statement, $\mathcal{F}_{|F}$ defines an algebraically integrable foliation on $X^0$ with $\mathcal{F}_{|X^0}$-leaves and combines with previous argument, we will get two projective subvariety on two fibre $F$ and $F'$ have a common point by theorem 3 (applies for two $\mathcal{F}_{|X_0}$-leaves which lie in $F$ and $F'$ respectedly), which is impossible because any of two fibre can't have a non-zero intersect. Hence, $dim T^0 = 0$ and by lemma 1, $X \cong \mathbb{P}^n$.
Conferences:
[Jie17] Jie Liu. Géométrie des variétés de Fano: Sous - Faisceaux du fibré tangent et diviseur fondomental. Thèse De Doctorat, Université Côte d'Azur.
[Ara06] Carolina Araujo. Rational curves of
minimal degree and characterizations of projective
spaces. Math. Ann., 335(4): 937 -- 951, 2006.
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