Canonical growth conditions associated to ample line bundles.(English)Zbl 1387.14036

In this extremely interesting paper, the author provides a new construction which associates to an ample (or big) line bundle $$L$$ on a projective manifold $$X$$ a canonical growth condition on the tangent space $$T_{p}X$$ of any given point $$p$$. Let us recall the main construction of the paper. Let $$(X,L)$$ be a polarized manifold and $$p \in X$$ a point. Pick local holomorphic coordinates $$z_{i}$$ centered at $$p$$, and choose a local trivialization of $$L$$ near $$p$$. Then any holomorphic section of $$L$$ (or more generally $$kL$$) can be written locally as a Taylor series $s = \sum a_{\alpha}z^{\alpha}.$ Let us denote by $$\text{ord}_{p}(s)$$ the order of vanishing of $$s$$ at $$p$$. Then leading order homogeneous part of $$s$$, which we will denote by $$s_{\mathrm{hom}}$$, is then given by $s_{\mathrm{hom}} := \sum_{|\alpha| = \text{ord}_{p}(s)} a_{\alpha}z^{\alpha},$ or if $$s \equiv 0$$, we let $$s_{\mathrm{hom}} \equiv 0$$. If $$\gamma(t)$$ is a smooth curve in $$\mathbb{C}^{n}$$ of the form $$\gamma(t) = tz_{0} +t^{2}h(t)$$, then one easily checks that $\lim_{t\rightarrow 0} \frac{s(\gamma(t))}{t^{\text{ord}_{p}(s)}} = s_{\mathrm{hom}}(z_{0}),$ which shows that $$s_{\mathrm{hom}}$$ in fact is a well-defined homogeneous holomorphic function on the tangent space $$T_{p}X$$. A different choice of trivialization would have the trivial effect of multiplying each $$s_{\mathrm{hom}}$$ by a fixed constant. Pic a smooth metric $$\phi$$ on $$L$$. This gives rise to supremum norms on each vector space $$H^{0}(X,kL)$$ simply by $||s||^{2}_{k\phi , \infty} := \text{sup} \{ |s(x)|^{2}e^{-k\phi} \}.$ Let $$B_{1}(kL, k\phi ) := \{s \in H^{0}(X,kL) : ||s||_{k \phi , \infty} \leq 1 \}$$ be the corresponding unit ball in $$H^{0}(X,kL)$$.
Definition 1. Let $\phi_{L,p} := \overset{*}{\text{sup}} \bigg\{ \frac{1}{k} \text{ln} |s_{\mathrm{hom}}|^{2} : s \in B_{1}(kL, k\phi ), k \in \mathbb{N} \bigg\}.$ Here $$*$$ means taking the upper semicontinuous regularization of the supremum.
One can show that $$\phi_{L,p}$$ is locally bounded from above, hence this function is a PSH function on $$T_{p}X$$. It is easy to show that if $$\phi^{'}$$ is some other smooth metric on $$L$$ and $$|\phi - \phi^{'}|< C$$, then $$|\phi_{L,p} - \phi^{'}_{L,p}| < C$$. Thus the growth condition $$\phi_{L,p} + \mathcal{O}(1)$$ on $$T_{p}X$$ is well-defined and depends only on the data $$X$$, $$L$$, and $$p$$. We call it the canonical growth condition of $$L$$ at $$p$$.
Now we present the main results of the paper.
Theorem A We have that $\int_{X} c_{1}(L)^{n} = (L)^{n} = \int_{T_{p}X}MA(\phi_{L,p}),$ where $$MA(\cdot)$$ denotes the Monge-Ampère measure.
Let us recall that for an ample line bundle $$L$$ and a point $$p \in X$$ the Seshadri constant is defined as $\varepsilon (X,L;p) = \text{inf}_{C} \frac{ L.C}{\text{mult}_{p}C},$ where the infimum is taken over all curves $$C$$ in $$X$$.
Theorem B. One has $\varepsilon(X,L;p) = \text{sup} \{ \lambda : \lambda \, \text{ln}(1 + |z|^{2}) \leq \phi_{L,p} + \mathcal{O}(1) \}.$
The next result of the paper is devoted to infinitesimal Newton-Okounkov bodies is a sense of A. Küronya and V. Lozovanu [Duke Math. J. 166, No. 7, 1349–1376 (2017; Zbl 1366.14012)].
Theorem C. The canonical growth condition $$\phi_{L,p} + \mathcal{O}(1)$$ completely determines all the infinitesimal Newton-Okounkov bodies $$\triangle(L, V_{\bullet})$$ of $$L$$ at $$p$$.
In order to formulate the last main result, we need to recall two definitions.
Definition 2. Let $$\omega_{0}$$ be a Kähler form on $$\mathbb{C}^{n}$$. We say that $$\omega_{0}$$ fits into $$(X,L)$$ if for any $$R>0$$ there exists a Kähler form $$\omega_{R}$$ on $$X$$ in $$c_{1}(L)$$ together with a Kähler embedding $$f_{R}$$ of the ball $$(B_{R},\omega_{0}|_{B_{R}})$$ into $$(X,\omega_{R})$$, where $$B_{R} = \{ |z|< R\} \subset \mathbb{C}^{n}$$. If the embeddings $$f_{R}$$ all can be chosen to map the origin to some fixed point $$p \in X$$, we say that $$\omega_{0}$$ fits into $$(X,L)$$ at $$p$$.
Definition 3. If $$u$$ and $$v$$ are two real-valued functions on $$\mathbb{C}^{n}$$, we say that $$u$$ grows faster than $$v$$ (and that $$v$$ grows slower than $$u$$) if $$u-v$$ is bounded from below and proper (for any constant $$C$$ there is an $$R > 0$$ such that $$u-v > C$$ on $$\{z : |z|>r\}$$).
Theorem D. Let $$\omega_{0} = dd^{c}\phi_{0}$$ be a Kähler form on $$\mathbb{C}^{n}$$. If for some isomorphism $$\mathbb{C}^{n} \simeq T_{p}X$$ we have that $$\phi_{0}$$ grows slower than $$\phi_{L,p}$$, then $$\omega_{0}$$ fits into $$(X,L)$$ at $$p$$.
As it is pointed out by the author, the above results can be generalized to the case when $$L$$ is big after suitable changes (i.e., in the case of Theorem B one needs to consider the so-called moving Seshadri constants).

MSC:

 14C20 Divisors, linear systems, invertible sheaves 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)

Zbl 1366.14012
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References:

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