arXiv 1711.034132017-11math.AG

完全交的切丛的稳定性与有效约束

Stability of tangent bundles of complete intersections and effective restriction

Jie Liu

备注：: The paper is significantly rewritten. More results on complete intersections in Hermitian symmetric spaces are included. Any comments are welcome

摘要：

对于$n\geq 3$，设$M$是一个$(n+r)$维不可约紧致型Hermite对称空间，$\mathcal{O}_M(1)$是$Pic(M)$的充分生成元.令$Y=H_1\cap\dots\cap H_r$是维数$n$的光滑完全相交，其中$H_i\in\vert \mathcal{O}_M(d_i)\vert$与$d_i\geq 2$。在$Y$上证明了扭全纯形式的一个消失定理。作为应用，我们证明了$Y$的切丛$T_Y$是稳定的。此外，如果$X$是$Y$中的次数为$d$的光滑超曲面，使得约束$Pic(Y)\rightarrow Pic(X)$是满射的，我们建立了一些关于$d$的有效结果，以保证约束$T_Y\vert_X$的稳定性.特别地，当$Y$是$\mathbb{P}^{n+1}$中的一般超曲面，$X$是$Y$中的一般光滑因子时，我们证明了除了一些著名的例子外，$T_Y\vert_X$是稳定的.我们还解决的情况下，皮卡德集团增加的限制。

英文摘要：

For $n\geq 3$, let $M$ be an $(n+r)$-dimensional irreducible Hermitian symmetric space of compact type and let $\mathcal{O}_M(1)$ be the ample generator of $Pic(M)$. Let $Y=H_1\cap\dots\cap H_r$ be a smooth complete intersection of dimension $n$ where $H_i\in\vert \mathcal{O}_M(d_i)\vert$ with $d_i\geq 2$. We prove a vanishing theorem for twisted holomorphic forms on $Y$. As an application, we show that the tangent bundle $T_Y$ of $Y$ is stable. Moreover, if $X$ is a smooth hypersurface of degree $d$ in $Y$ such that the restriction $Pic(Y)\rightarrow Pic(X)$ is surjective, we establish some effective results for $d$ to guarantee the stability of the restriction $T_Y\vert_X$. In particular, if $Y$ is a general hypersurface in $\mathbb{P}^{n+1}$ and $X$ is general smooth divisor in $Y$, we show that $T_Y\vert_X$ is stable except for some well-known examples. We also address the cases where the Picard group increases by restriction.

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摘要：

英文摘要：

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