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).


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


Zbl 1366.14012
Full Text: DOI arXiv Euclid


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