We give explicit estimates for the volume of the Green–Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green–Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one.
On donne des estimées explicites pour le volume des différentielles de jets de Green–Griffiths à tout ordre sur une compactification toroïdale d’un quotient de la boule. Pour ce faire, on détermine tout d’abord l’ordre de croissance des différentielles de jets de Green–Griffiths logarithmiques sur ces objets, en utilisant une déformation naturelle de l’espace des jets logarithmiques d’un ordre fixé, vers un fibré projectivisé à poids adéquat. Ensuite, on estime la croissance du nombre de conditions d’annulation au bord qu’une différentielle de jets logarithmique doit satisfaire pour être une différentielle de jets standard.
Accepted:
Published online:
Keywords: Jet differentials, Ball quotients
Mot clés : Différentielles de jets, quotients de la boule
Cadorel, Benoît 1
@article{AIF_2020__70_6_2331_0, author = {Cadorel, Beno{\^\i}t}, title = {Jet differentials on toroidal compactifications of ball quotients}, journal = {Annales de l'Institut Fourier}, pages = {2331--2359}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {6}, year = {2020}, doi = {10.5802/aif.3356}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3356/} }
TY - JOUR AU - Cadorel, Benoît TI - Jet differentials on toroidal compactifications of ball quotients JO - Annales de l'Institut Fourier PY - 2020 SP - 2331 EP - 2359 VL - 70 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3356/ DO - 10.5802/aif.3356 LA - en ID - AIF_2020__70_6_2331_0 ER -
%0 Journal Article %A Cadorel, Benoît %T Jet differentials on toroidal compactifications of ball quotients %J Annales de l'Institut Fourier %D 2020 %P 2331-2359 %V 70 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3356/ %R 10.5802/aif.3356 %G en %F AIF_2020__70_6_2331_0
Cadorel, Benoît. Jet differentials on toroidal compactifications of ball quotients. Annales de l'Institut Fourier, Volume 70 (2020) no. 6, pp. 2331-2359. doi : 10.5802/aif.3356. https://aif.centre-mersenne.org/articles/10.5802/aif.3356/
[1] Cohomological Study of Weighted Projective Spaces, Algebraic Geometry (Sertoz, Sinan, ed.) (Lecture Notes in Pure and Applied Mathematics), Volume 193, CRC Press, 1997 | MR | Zbl
[2] Smooth compactifications of locally symmetric varieties, Cambridge University Press, 2010, x+230 pages (with the collaboration of Peter Scholze) | DOI | Zbl
[3] The Kodaira dimension of complex hyperbolic manifolds with cusps, Compos. Math., Volume 154 (2018) no. 3, pp. 549-564 | DOI | MR | Zbl
[4] Poincaré–Birkhoff–Witt theorem for quadratic algebras of Koszul type, J. Algebra, Volume 181 (1996) no. 2, pp. 315-328 | DOI | Zbl
[5] Symmetric differentials on complex hyperbolic manifolds with cusps (2016) (https://arxiv.org/abs/1606.05470, to appear in J. Differ. Geom.)
[6] Variétés faiblement spéciales à courbes entières dégénérées, Compos. Math., Volume 143 (2007) no. 1, pp. 95-111 | DOI | MR | Zbl
[7] Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier, Volume 65 (2015) no. 2, pp. 835-861 | DOI | Numdam | MR | Zbl
[8] Holomorphic Morse inequalities and the Green–Griffiths–Lang conjecture, Pure Appl. Math. Q., Volume 7 (2011) no. 4, pp. 1165-1207 | DOI | MR | Zbl
[9] Hyperbolic algebraic varieties and holomorphic differential equations, Acta Math. Vietnam., Volume 37 (2012) no. 4, pp. 441-512 | MR | Zbl
[10] Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981) (Lecture Notes in Mathematics), Volume 956, Springer, 1982, pp. 34-71 | DOI | MR | Zbl
[11] Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2, Springer, 1998, xiv+470 pages | MR | Zbl
[12] Positivity in algebraic geometry. I Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48, Springer, 2004, xviii+387 pages | Zbl
[13] Positivity in algebraic geometry. II Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 49, Springer, 2004, xviii+385 pages | MR | Zbl
[14] Projective algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume, Perspectives in analysis, geometry, and topology (Progress in Mathematics), Volume 296, Birkhäuser, 2012, pp. 331-354 | DOI | MR | Zbl
[15] Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math., Volume 42 (1977), pp. 239-272 | DOI | MR | Zbl
[16] Weighted Projective Embeddings, Stability of Orbifolds, and Constant Scarlar Curvature Kähler Metrics, J. Differ. Geom., Volume 88 (2011) no. 1, pp. 109-159 | DOI | Zbl
[17] Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford, -Theory, Volume 18 (1999) no. 1, pp. 33-76 | DOI | Zbl
Cited by Sources: