×

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)

Citations:

Zbl 1366.14012
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] D. Anderson,Okounkov bodies and toric degenerations, Math. Ann.356(2013), 1183-1202. · Zbl 1273.14104
[2] E. Bedford and B. A. Taylor,A new capacity for plurisubharmonic functions, Acta Math.149(1982), 1-40. · Zbl 0547.32012
[3] R. J. Berman,Bergman kernels and equilibrium measures for line bundles over projective manifolds, Amer. J. Math.131(2009), 1485-1524. · Zbl 1191.32008
[4] R. Berman and S. Boucksom,Growth of balls of holomorphic sections and energy at equilibrium, Invent. Math.181(2010), 337-394. · Zbl 1208.32020
[5] S. Boucksom,Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. Éc. Norm. Supér. (4)37(2004), 45-76. · Zbl 1054.32010
[6] S. Boucksom,Corps d’Okounkov (d’après Okounkov, Lazarsfeld-Mustaţă et Kaveh-Khovanskii), Astérisque361(2014), Exp. No. 1059, 1-41. · Zbl 1365.14059
[7] S. Boucksom, J.-P. Demailly, M. Păun, and T. Peternell,The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom.22(2013), 201-248. · Zbl 1267.32017
[8] S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi,Monge-Ampère equations in big cohomology classes, Acta Math.205(2010), 199-262. · Zbl 1213.32025
[9] S. Boucksom, C. Favre, and M. Jonsson,Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom.18(2009), 279-308. · Zbl 1162.14003
[10] J.-P. Demailly,Regularization of closed positive currents and intersection theory, J. Algebraic Geom.1(1992), 361-409. · Zbl 0777.32016
[11] J.-P. Demailly, “Singular Hermitian metrics on positive line bundles” inComplex Algebraic Varieties (Bayreuth, 1990), Lecture Notes in Math.1507, Springer, Berlin, 1992, 87-104.
[12] M. Harada and K. Kaveh,Integrable systems, toric degenerations and Okounkov bodies, Invent. Math.202(2015), 927-985. · Zbl 1348.14122
[13] T. Hisamoto,Restricted Bergman kernel asymptotics, Trans. Amer. Math. Soc.364, no. 7 (2012), 3585-3607. · Zbl 1273.32006
[14] T. Hisamoto,On the volume of graded linear series and Monge-Ampère mass, Math. Z.275(2013), 233-243. · Zbl 1277.32019
[15] A. Ito,Okounkov bodies and Seshadri constants, Adv. Math.241(2013), 246-262. · Zbl 1282.14013
[16] K. Kaveh,Toric degenerations and symplectic geometry of smooth projective varieties, preprint,arXiv:1508.00316v3[math.SG]. · Zbl 07053666
[17] K. Kaveh and A. G. Khovanskii, “Algebraic equations and convex bodies” inPerspectives in Analysis, Geometry, and Topology, Progr. Math.296, Birkhäuser/Springer, New York, 2012, 263-282. · Zbl 1316.52011
[18] K. Kaveh and A. G. Khovanskii,Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2)176(2012), 925-978. · Zbl 1270.14022
[19] C. O. Kiselman,The partial Legendre transformation for plurisubharmonic functions, Invent. Math.49(1978), 137-148. · Zbl 0378.32010
[20] A. Küronya and V. Lozovanu,Infinitesimal Newton-Okounkov bodies and jet separation, Duke Math. J.166(2017), 1349-1376. · Zbl 1366.14012
[21] A. Küronya and V. Lozovanu,Positivity of line bundles and Newton-Okounkov bodies, preprint,arXiv:1506.06525v1[math.AG].
[22] R. Lazarsfeld,Positivity in Algebraic Geometry, I: Classical Setting: Line Bundles and Linear Series, Ergeb. Math. Grenzgeb.48, Springer, Berlin, 2004. · Zbl 1093.14501
[23] R. Lazarsfeld and M. Mustaţă,Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4)42(2009), 783-835. · Zbl 1182.14004
[24] D. McDuff and L. Polterovich,Symplectic packings and algebraic geometry, with an appendix by Y. Karshon, Invent. Math.115(1994), 405-434. · Zbl 0833.53028
[25] M. Nakamaye,Base loci of linear series are numerically determined, Trans. Amer. Math. Soc.355, no. 2 (2003), 551-566. · Zbl 1017.14017
[26] A. Okounkov,Brunn-Minkowski inequality for multiplicities, Invent. Math.125(1996), 405-411. · Zbl 0893.52004
[27] A. Okounkov, “Why would multiplicities be log-concave?” inThe Orbit Method in Geometry and Physics (Marseille, 2000), Progr. Math.213, Birkhäuser, Boston, 2003, 329-347. · Zbl 1063.22024
[28] R. Richberg,Stetige streng pseudokonvexe Funktionen, Math. Ann.175(1968), 257-286. · Zbl 0153.15401
[29] J. Ross and D. Witt Nyström,Homogeneous Monge-Ampère equations and canonical tubular neighbourhoods in Kähler geometry, preprint,arXiv:1403.3282v2[math.CV].
[30] W.-D. Ruan, “Lagrangian torus fibration of quintic hypersurfaces, I: Fermat quintic case” inWinter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, Mass., 1999), AMS/IP Stud. Adv. Math.23, Amer. Math. Soc., Providence, 2001, 297-332.
[31] Y. T. Siu,Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math.27(1974), 53-156. · Zbl 0289.32003
[32] D. Witt Nyström,Transforming metrics on a line bundle to the Okounkov body, Ann. Sci. Éc. Norm. Supér. (4)47(2014), 1111-1161. · Zbl 1329.14025
[33] D. Witt Nyström,Canonical growth conditions associated to ample line bundles, preprint,arXiv:1509.05528v4[math.AG].
[34] D. Witt Nyström,Okounkov bodies and embeddings of torus-invariant Kähler balls, preprint,arXiv:1510.00510v3[math.AG].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.