PomeD45322F36C36We now remove the assumption that $X$ carries a complete flag. A curve always does, since any point completes the flag. In dimension at least two, the exceptional divisor of the blowup of a point is a projective space and carries a complete flag of linear subspaces, while Hartogs extension identifies the sections on the blowup with those on $X$. The growth formula on the blowup therefore counts the original section spaces.
Lemma 19.9.15.744C8C462232 Every connected compact complex manifold $X$ of dimension $n\geq 1$ admits a smooth compact modification $\pi :\widetilde X\to X$ such that $\widetilde X$ carries a complete flag and, for every holomorphic line bundle $L$ on $X$ and every $m\geq 0$, pullback is an isomorphism
These isomorphisms are compatible with products.
Proof. If $n=1$, take $\pi $ to be the identity and any point as the last member of the flag; the remaining assertions are then immediate. If $n\geq 2$, blow up a point $p\in X$. The local description of the blowup of $\mathbb C^ n$ at the origin shows that $\widetilde X$ is smooth and that its exceptional divisor is $E\simeq \mathbb P^{n-1}$. A nested sequence of linear subspaces of $E$, together with $\widetilde X$, gives a flag
Each member is smooth, compact and connected, and is a prime divisor in the preceding member. The modification is an isomorphism off $E$.
Let $s$ be a section of $\pi ^*L^ m$. Off $E$, it gives a section of $L^ m$ on $X\setminus \{ p\} $. In a trivialization of $L^ m$ on a coordinate ball centered at $p$, its coefficient is holomorphic on the punctured ball, and by Hartogs' extension theorem, valid since $n\geq 2$, it extends holomorphically across $p$. The extension glues with the section away from $p$ to a section of $L^ m$ on $X$. Its pullback agrees with $s$ off $E$, hence everywhere by the identity theorem. This proves surjectivity; injectivity and compatibility with products follow because pullback is an isomorphism on the dense open set $\widetilde X\setminus E$. $\square$
Given a graded subalgebra $W_\bullet $ of the section ring of $L$, choose a complete flag on the modification of lemma 19.9.15 and apply the constructions of subsection 19.9.5 on $\widetilde X$ to the pulled-back algebra $\widetilde W_ m=\pi ^*W_ m\subseteq H^0(\widetilde X,\pi ^*L^ m)$. The shape of the resulting body may depend on the point and on the flag, but its dimension and its normalized count are determined by $W_\bullet $.
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
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. $\square$
Corollary 19.9.17.EECB2B1DF1A4 For every holomorphic line bundle $L$ on a connected compact complex manifold $X$ of dimension $n$,
If $X$ is not Moishezon, then, for each $L$, either all positive-degree section spaces of $L$ vanish, or there is an integer $d<n$ for which
In particular $m^{-n}h^0(X,L^ m)\to 0$. These conclusions apply when $X$ fails the bounded mass property.
Proof. If all positive-degree section spaces vanish, there is nothing to prove. Otherwise, apply the theorem to the full section ring. Since $W_ m=0$ for $e\nmid m$, the limit gives $h^0(X,L^ m)=O(m^ d)$ for all degrees, with $d\leq n$. If $d=n$, the section-ratio field contains $n$ algebraically independent meromorphic functions on $X$, so $X$ is Moishezon. Thus $d<n$ when $X$ is not Moishezon. Finally, by Moishezon's theorem a Moishezon manifold admits a modification from a projective, hence Kähler, manifold, so it has the bounded mass property by lemma 19.9.12. The contrapositive gives the last assertion. $\square$
Since the multiplier-ideal section spaces $\mathcal H_ m(u)$ of subsection 19.8.1 are contained in $H^0(X,L^ m)$, the same upper bounds hold for them. Identifying weighted section spaces on a modification, however, requires a correction by the Jacobian; lemma 5.43.22 gives the corresponding formula involving the relative canonical bundle.
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.