论文标题
截面环的度量结构
On the metric structure of section ring
论文作者
论文摘要
本文的主要目的是研究一个射频歧管，并在其上有足够的线条捆绑包在相关部分环上的度量和代数结构之间的关系。 更确切地说，我们证明一旦核算内核，就$ l^2 $ norms和诱导的Hermitian Tensor产品规范而言，截面环的乘法运算符将变成近似等轴测图（达到归一化）。我们还表明，类似的结果适用于$ l^1 $和$ l^{\ infty} $ - 如果而不是Hermitian Tensor产品规范，我们将分别考虑由$ l^1 $和$ L^1 $和$ L^{\ f^{\ infty} $ norms引起的投影和注射性张量规范。 然后，我们表明与连续Plurisubharmonic相关的$ l^2 $ norms实际上以这种类型的多重性属性为特征。使用它，我们将Phong-Sturm的定理从较弱的Fubini-study收敛水平到更强的规范等价水平的量化定理。
The main goal of this article is to study for a projective manifold and an ample line bundle over it the relation between metric and algebraic structures on the associated section ring. More precisely, we prove that once the kernel is factored out, the multiplication operator of the section ring becomes an approximate isometry (up to normalization) with respect to the $L^2$-norms and the induced Hermitian tensor product norm. We also show that the analogous result holds for the $L^1$ and $L^{\infty}$-norms if instead of the Hermitian tensor product norm, we consider the projective and injective tensor norms induced by $L^1$ and $L^{\infty}$-norms respectively. Then we show that $L^2$-norms associated with continuous plurisubharmonic metrics are actually characterized by the multiplicativity properties of this type. Using this, we refine the theorem of Phong-Sturm about quantization of Mabuchi geodesics from the weaker level of Fubini-Study convergence to the stronger level of norm equivalences.
