随机Kodaira嵌入的期望质心
Expected centre of mass of the random Kodaira embedding
摘要：
设$X \subset \mathbb{P}^{N-1}$是光滑射影簇.对于诱导由环境线性作用给出的嵌入$g \cdot X \subset \mathbb{P}^{N-1}$的每个$g \in SL (N , \mathbb{C})$，我们可以关联称为质心的矩阵$\barμ_X (g)$，其非线性地依赖于$g$。对于由Haar测度和高斯酉系综诱导的$SL (N , \mathbb{C})$上的概率测度，证明了对任意光滑射影簇，质心的期望是单位矩阵的常数倍.
英文摘要：
Let $X \subset \mathbb{P}^{N-1}$ be a smooth projective variety. To each $g \in SL (N , \mathbb{C})$ which induces the embedding $g \cdot X \subset \mathbb{P}^{N-1}$ given by the ambient linear action we can associate a matrix $\barμ_X (g)$ called the centre of mass, which depends nonlinearly on $g$. With respect to the probability measure on $SL (N , \mathbb{C})$ induced by the Haar measure and the Gaussian unitary ensemble, we prove that the expectation of the centre of mass is a constant multiple of the identity matrix for any smooth projective variety.
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