Pome

Theorem 19.9.161EBEF54DB76D

Theorem 19.9.16.1EBEF54DB76D Assume that some positive-degree piece of $W_\bullet $ is nonzero. There are an integer $0\leq d\leq n$, a degree period $e\geq 1$, and a number $V(W_\bullet )>0$ such that

\[ \lim _{q\to \infty }\frac{\dim W_{eq}}{(eq)^ d}=V(W_\bullet ), \qquad W_ m=0\quad \hbox{if }e\nmid m. \]

Here $d$ is the transcendence degree of the section-ratio field $K(W_\bullet )$, $e$ is the greatest common divisor of the degrees $m$ with $W_ m\neq 0$, and $V(W_\bullet )$ is the intrinsic lattice volume of every body constructed on such a modification. In particular, these three quantities do not depend on the choice of the modification or the flag.

Proof. Pullback preserves the graded dimensions and the degrees with nonzero sections. Applying theorem 19.9.3, theorem 19.9.9 on $\widetilde X$ gives the displayed limit and the positivity of the lattice volume. Pullback also identifies the section-ratio fields, since their elements are the corresponding ratios and pullback is injective on the dense set where $\pi $ is an isomorphism. Thus proposition 19.9.10 identifies $d$ with the transcendence degree of the original field. The unchanged degrees with nonzero sections determine $e$, and the limit of the unchanged dimensions determines $V(W_\bullet )$. None of the three depends on the auxiliary choices. $\square$

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