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On the Kodaira dimension of maximal orders. (arXiv:1611.10278v2 [math.AG] UPDATED)

来源于:arXiv
Let ${k}$ be an algebraically closed field of characteristic zero and let ${K}$ be a field finitely generated over ${k}$. Let $\Sigma$ be a central simple ${K}$-algebra, $X$ a normal projective model of ${K}$ and $\Lambda$ a sheaf of maximal $\mathcal{O}_X$-orders in $\Sigma$. There is a ramification ${Q}$-divisor $\Delta$ on $X$, which is related to the canonical bimodule $\omega_\Lambda$ by an adjunction formula, and only depends on the class of $\Sigma$ in the Brauer group of ${K}$. When the numerical abundance conjecture holds true, or when $\Sigma$ is a division algebra, we show that the Gelfand-Kirillov dimension (or GK dimension) of the canonical ring of $\Lambda$ is one more than the Iitaka dimension (or D-dimension) of the log pair $(X,\Delta)$. In the case that $\Sigma$ is a division algebra, we further show that this GK dimension is also one more than the transcendence degree of the division algebra of degree zero fractions of the canonical ring of $\Lambda$. We prove that t 查看全文>>