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arXiv 1902.05348math.DGmath.AG
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射影流形的第二基本型的刚性

The rigidity on the second fundamental form of projective manifolds

Ping Li

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10 pages, final version to appear in Manuscripta Mathematica
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摘要：

设$M$是$\mathbb{P}^{n+r}$中的复$n$维射影流形，具有常全纯截面曲率$1$的Fubini-Study度量，$σ$是它的第二基本型，$\underline{|σ|}^2$是$M$上$σ$的平方长度的平均值.我们推导出一个公式$\underline{|σ|}^2$和分类时，$\underline{|σ|}^2\leq2n$。我们提出了几个应用这些结果。第一个应用是证实Loi和Zedda的一个猜想，该猜想用$σ$的$L^2$ -范数刻画了线性子空间和二次曲面.第二个应用是将Cheng解决Oguie旧猜想的结果从逐点情况改进为平均情况。第三个应用是给出了$\underline{|σ|}^2$上的最优第二间隙值，它可以看作是单位球面上极小子流形上的最优第二间隙值的一个复类比。

英文摘要：

Let $M$ be a complex $n$-dimensional projective manifold in $\mathbb{P}^{n+r}$ endowed with the Fubini-Study metric of constant holomorphic sectional curvature $1$, $σ$ its second fundamental form, and $\underline{|σ|}^2$ the mean value of the squared length of $σ$ on $M$. We derive a formula for $\underline{|σ|}^2$ and classify them when $\underline{|σ|}^2\leq2n$. We present several applications to these results. The first application is to confirm a conjecture of Loi and Zedda, which characterizes the linear subspace and the quadric in terms of the $L^2$-norm of $σ$. The second application is to improve a result of Cheng solving an old conjecture of Oguie from pointwise case to mean case. The third application is to give an optimal second gap value on $\underline{|σ|}^2$, which can be viewed as a complex analog to those on minimal submanifolds in the unit spheres.

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