On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces

Juanyong Wang

doi:10.5802/afst.1690

On the Iitaka Conjecture

C_{n, m}

for Kähler Fibre Spaces

Juanyong Wang ¹

¹ Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 4, pp. 813-897

Abstract (VO)
Résumé

By applying the positivity theorem of direct images and a pluricanonical version of the structure theorem on the cohomology jumping loci à la Green–Lazarsfeld–Simpson, we show that the klt Kähler version of the Iitaka conjecture $C_{n, m}$ (Ueno, 1975) for $f : X \to Y$ (surjective morphism between compact Kähler manifolds with connected general fibre) holds true when the determinant of the direct image of some power of the relative canonical bundle is big on $Y$ or when $Y$ is a complex torus. These generalize the corresponding results of Viehweg (1983) and of Cao-Păun (2017) respectively. We further generalize the later case to the geometric orbifold setting, i.e. prove that $C_{n, m}^{orbifold}$ (Campana, 2004) holds when $Y$ is a complex torus.

En appliquant la positivité des images directes et une version pluricanonique du théorème de structure des lieux de saut cohomologique à la Green–Lazarsfeld–Simpson, nous démontrons que la version klt kählérienne de la conjecture d’Iitaka $C_{n, m}$ (Ueno, 1975) pour $f : X \to Y$ (morphisme surjectif entre variétés kählériennes compactes à fibre générale connexe) est vraie si le déterminant de l’image directe d’une certaine puissance du fibré canonique relative est gros sur $Y$ ou si $Y$ est un tore complexe. Ceci généralisent les résultats correspondants de Viehweg (1983) et de Cao-Păun (2017) respectivement. De plus nous généralisons le deuxième résultat ci-dessus au cadre des orbifoldes géométriques, c-à-d., nous démontrons que $C_{n, m}^{orbifold}$ (Campana, 2004) est vraie quand $Y$ est un tore complexe.

Reçu le : 2019-07-16
Accepté le : 2020-01-24
Publié le : 2021-12-06

DOI : 10.5802/afst.1690

Affiliations des auteurs :

Juanyong Wang ¹

¹ Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Juanyong Wang},
     title = {On the {Iitaka} {Conjecture~}$C_{n,m}$ for {K\"ahler} {Fibre} {Spaces}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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     year = {2021},
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Juanyong Wang. On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 4, pp. 813-897. doi: 10.5802/afst.1690

Bibliographie
Cité par

[1] Michael Artin Néron Models, Arithmetic Geometry, Springer (1986), pp. 213-230 | DOI | Zbl

[2] Constantin Bănică; Octavian Stănăşilă Algebraic Methods in the Global Theory of Complex Spaces, John Wiley & Sons, 1976 (revised English version of Metode algebrice în teoria globală a spaţiilor complexe, Etitura academiei, Bucureşti, 1974)

[3] Bo Berndtsson Curvature of Vector Bundles Associated to Holomorphic Fibrations, Ann. Math., Volume 169 (2009) no. 2, pp. 531-560 | DOI | MR | Zbl

[4] Bo Berndtsson The Openness Conjecture for Plurisubharmonic Functions (2013) (https://arxiv.org/abs/1305.5781)

[5] Bo Berndtsson; Mihai Păun Bergman Kernels and the Pseudoeffectivity of Relative Canonical Bundles, Duke Math. J., Volume 145 (2008) no. 2, pp. 341-378 | DOI | MR | Zbl

[6] Bo Berndtsson; Mihai Păun Bergman Kernels and the Subadjunction (2010) (https://arxiv.org/abs/1002.4145)

[7] Christina Birkenhake; Herberte Lang Complex Abelian Varieties, Grundlehren der Mathematischen Wissenschaften, 302, Springer, 2004 | DOI

[8] Zbigniew Błocki Suita Conjecture and the Ohsawa-Takegoshi Extension Theorem, Invent. Math., Volume 193 (2013) no. 1, pp. 149-158 | DOI | MR | Zbl

[9] Sébastien Boucksom Divisorial Zariski Decomposition on Compact Complex Manifolds, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 4, pp. 45-67 | DOI | MR | Zbl

[10] Sébastien Boucksom Singularities of Plurisubharmonic Functions and Multiplier Ideals, 2016 (Notes of Course, http://sebastien.boucksom.perso.math.cnrs.fr/notes/L2.pdf)

[11] Sébastien Boucksom; Jean-Pierre Demailly; Miahi Păun; Thomas Peternell The Pseudo-effective Cone of a Compact Kähler Manifold and Varieties of Negative Kodaira Dimension, J. Algebr. Geom., Volume 22 (2013) no. 2, pp. 201-248 | DOI | Zbl

[12] Sébastien Boucksom; Tommaso de Fernex; Charles Favre The Volume of an Isolated Singularity, Duke Math. J., Volume 161 (2012) no. 8, pp. 1455-1520 | MR | Zbl

[13] Sébastien Boucksom; Mattias Jonsson Tropical and non-Archimedean Limits of Degenerating Families of Volume Forms, J. Éc. Polytech., Math., Volume 4 (2017), pp. 87-139 | DOI | MR | Numdam | Zbl

[14] Nero Budur Unitary Local Systems, Multiplier Ideals, and Polynomial Periodicity of Hodge Numbers, Adv. Math., Volume 221 (2009) no. 1, pp. 217-250 | DOI | MR | Zbl

[15] Nero Budur; Botong Wang Cohomology Jump Loci of Quasi-projective Varieties, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015) no. 1, pp. 227-236 | DOI | MR | Zbl

[16] Nero Budur; Botong Wang Cohomology Jump Loci of Quasi-compact Kähler Manifolds (2017) (https://arxiv.org/abs/1702.02186)

[17] Frédéric Campana Orbifolds, Special Varieties and Classification Theory, Ann. Inst. Fourier, Volume 54 (2004) no. 3, pp. 499-665 | Numdam | MR | Zbl | DOI

[18] Frédéric Campana Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes (2009) (Prépublication IECN https://hal.archives-ouvertes.fr/hal-00356763)

[19] Frédéric Campana; Andreas Höring; Thomas Peternell Abundance for Kähler Threefolds, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 4, pp. 971-1025 | DOI | Zbl

[20] Frédéric Campana; Vincent Koziarz; Mihai Păun Numerical Character of the Effectivity of Adjoint Line Bundles, Ann. Inst. Fourier, Volume 62 (2012) no. 1, pp. 107-119 | DOI | Numdam | Zbl | MR

[21] Frédéric Campana; Thomas Peternell Geometric Stability of the Cotangent Bundle and the Universal Cover of a Projective Manifold, Bull. Soc. Math. Fr., Volume 139 (2011) no. 1, pp. 41-74 | DOI | MR | Numdam | Zbl

[22] Junyan Cao On the Approximation of Kähler Manifolds by Algebraic Varieties, Math. Ann., Volume 363 (2015) no. 1-2, pp. 393-422 | MR | Zbl

[23] Junyan Cao Ohsawa-Takegoshi Extension Theorem for Compact Kähler Manifolds and Applications, Complex and Symplectic Geometry (Springer INdAM Series), Volume 21, Springer (2017), pp. 19-38 | Zbl | MR

[24] Junyan Cao; Andreas Höring A Decomposition Theorem for Projective Manifolds with Nef Anticanonical Bundle (2017) (to appear in J. Algebr. Geom., https://arxiv.org/abs/1706.08814)

[25] Junyan Cao; Mihai Păun Kodaira Dimension of Algebraic Fibre Space over Abelian Varieties, Invent. Math., Volume 207 (2017) no. 1, pp. 345-387 | Zbl

[26] Henri Cartan Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes, Herman, éditeur des sciences et des arts, 1961

[27] Olivier Debarre Tores et variétés abéliennes complexes, Contributions in Mathematical and Computational Sciences, 6, Société Mathématique de France; EDP Sciences, 1999

[28] Olivier Debarre Higher-Dimensional Algebraic Geometry, Universitext, Springer, 2001 | DOI

[29] Jean-Pierre Demailly Analytic Methods in Algebraic Geometry, Surveys of Modern Mathematics, 1, Higher Education Press; International Press, 2010

[30] Jean-Pierre Demailly Complex Analytic and Differential Geometry, OpenContent Book https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf, 2012

[31] Jean-Pierre Demailly Extension of Holomorphic Functions Defined on non Reduced Analytic Subvarieties (2015) (https://arxiv.org/abs/1510.05230)

[32] Jean-Pierre Demailly; Lawrence Ein; Robert Lazarsfeld A Subadditivity Property of Multiplier Ideals, Mich. Math. J., Volume 48 (2000) no. 1, pp. 137-156 | MR | Zbl

[33] Fusheng Deng; Zhiwei Wang; Liyou Zhang; Xiangyu Zhou New Characterizations of Plurisubharmonic Functions and Positivity of Direct Image Sheaves (2018) (https://arxiv.org/abs/1809.10371)

[34] Ya Deng Applications of the Ohsawa–Takegoshi Extension Theorem to Direct Image Problems (2017) (https://arxiv.org/abs/1703.07279)

[35] Hélène Esnault Classification des variétés de dimension $3$ et plus, Séminaire Bourbaki n°23 (1981) (Lecture Notes in Mathematics), Volume 901, Springer (1981) (exposé 568) | DOI | Numdam | Zbl

[36] Osamu Fujino Notes on the Weak Positivity Theorem, Algebraic Varieties and Automorphism Groups, July 7-11, 2014 (Advances Studies in Pure Mathematics), Volume 75, Mathematical Society of Japan (2017), pp. 73-118 | DOI | MR | Zbl

[37] Takao Fujita On Kähler Fibre Spaces over Curves, J. Math. Soc. Japan, Volume 30 (1978) no. 4, pp. 779-794 | MR | Zbl

[38] William Fulton Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2, Springer, 1984 | DOI

[39] Hans Grauert; Reinhold Remmert Plurisubharmonische Funktionen in komplexen Räumen, Math. Z., Volume 65 (1956), pp. 175-194 | DOI | Zbl

[40] Mark Green; Phillip Griffiths Positivity of Vector Bundles and Hodge Theory (2018) (https://arxiv.org/abs/1803.07405)

[41] Mark Green; Robert Lazarsfeld Deformation Theory, Generic Vanishing Theorems, and some Conjectures of Enriques, Cantanese and Beauville, Invent. Math., Volume 90 (1987) no. 2, pp. 389-407 | DOI | Zbl

[42] Mark Green; Robert Lazarsfeld Higher Obstructions to Deforming Cohomology Groups of Line Bundles, J. Am. Math. Soc., Volume 4 (1991) no. 1, pp. 87-103 | DOI | MR | Zbl

[43] Qi’an Guan; Xiangyu Zhou A Proof of Demailly’s Strong Openness Conjecture, Ann. Math., Volume 182 (2015) no. 2, pp. 605-616 | DOI | MR | Zbl

[44] Qi’an Guan; Xiangyu Zhou A Solution of an $L^{2}$ Extension Problem with an Optimal Estimate and Applications, Ann. Math., Volume 181 (2015) no. 3, pp. 1139-1208 | DOI | MR | Zbl

[45] Christopher Hacon; Mihnea Popa; Christian Schnell Algebraic Fibre Spaces over Abelian Varieties: Around a Recent Theorem by Cao and Păun, Local and Global Methods in Algebraic Geometry (Contemporary Mathematics), Volume 712, American Mathematical Society (2018), pp. 143-195 | Zbl

[46] Heisuke Hironaka Flattening Theorem in Complex Analytic Geometry, Am. J. Math., Volume 97 (1975) no. 2, pp. 503-547 | DOI | Zbl | MR

[47] Andreas Höring Positivity of Direct Image Sheaves. A Geometric Point of View, Enseign. Math., Volume 56 (2010) no. 1-2, pp. 87-142 | DOI | Zbl | MR

[48] Andreas Höring; Thomas Peternell Minimal Models for Kähler Threefolds, Invent. Math., Volume 203 (2016) no. 1, pp. 217-264 | DOI | Zbl

[49] Yujiro Kawamata Characterization of Abelian Varieties, Compos. Math., Volume 43 (1981) no. 2, pp. 253-276 | MR | Numdam | Zbl

[50] Yujiro Kawamata Kodaira Dimension of Algebraic Fibre Spaces over Curves, Invent. Math., Volume 66 (1982) no. 1, pp. 57-71 | DOI | Zbl

[51] Yujiro Kawamata Minimal Models and the Kodaira Dimension of Algebraic Fibre Spaces, J. Reine Angew. Math., Volume 1985 (1985) no. 363, pp. 1-46 | Zbl

[52] Yujiro Kawamata On the Abundance Theorem in the Case of Numerical Kodaira Dimension Zero, Am. J. Math., Volume 135 (2013) no. 1, pp. 115-124 | DOI | MR | Zbl

[53] Yujiro Kawamata; Katsumi Matsuda; Kenji Matsuki Introduction to the Minimal Model Problem, Algebraic Geometry, Sendai, 1985 (Advanced Studies in Pure Mathematics), Volume 10, Mathematical Society of Japan (1987), pp. 283-360 | DOI | MR | Zbl

[54] Yujiro Kawamata; Eckart Viehweg On a Characterization of an Abelian Variety in the Classification Theory of Algebraic Varieties, Compos. Math., Volume 41 (1980) no. 3, pp. 355-359 | MR | Numdam | Zbl

[55] Shoshichi Kobayashi Differential Geometry of Complex Vector Bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, 1987 | DOI

[56] János Kollár Singularities of Pairs, Algebraic Geometry Santa Cruz 1995 (Proceedings of Symposia in Pure Mathematics), Volume 62, American Mathematical Society (1997), pp. 221-287 | DOI | MR | Zbl

[57] János Kollár; Shigefumi Mori Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998 | DOI

[58] George Rushing Kumpf; Finn Faye Knudsen; David Bryant Mumford; Bernard Saint-Donat Toroidal Embeddings, Lecture Notes in Mathematics, 339, Springer, 1973

[59] Robert Lazarsfeld Positivity in Algebraic Geometry I & II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48 & 49, Springer, 2004

[60] John Lesieutre A Pathology of Asymptotic Multiplicity in the Relative Setting, Math. Res. Lett., Volume 23 (2016) no. 5, pp. 1433-1451 | DOI | MR | Zbl

[61] Hideyuki Matsumura Commutative Algebra, Mathematics Lecture Note Series, W. A. Benjamin, Inc., 1970

[62] Hideyuki Matsumura Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1989

[63] James Stuart Milne Abelian Varieties, Arithmetic Geometry, Springer (1986), pp. 103-150 | DOI

[64] Noboru Nakayama Zariski-decomposition and Abundance, MSJ Memoirs, Mathematical Society of Japan, 2004

[65] Mihai Păun Singular Hermitian Metrics and Positivity of Direct Images of Pluricanonical Bundles, 2016 (Survey https://arxiv.org/abs/1606.00174) | Zbl

[66] Mihai Păun; Shigeharu Takayama Positivity of Twisted Relative Pluricanonical Bundles and Their Direct Images, J. Algebr. Geom., Volume 27 (2018) no. 2, pp. 211-272 | DOI | MR | Zbl

[67] Jean-Pierre Ramis; Gabriel Ruget Complexe dualisant et théorèmes de dualité en géométrie analytique complexe, Publ. Math., Inst. Hautes Étud. Sci., Volume 38 (1970), pp. 77-91 | Numdam | Zbl | DOI

[68] Hossein Raufi Singular Hermitian Metrics on Holomorphic Vector Bundles, Ark. Mat., Volume 53 (2015) no. 2, pp. 359-382 | DOI | MR | Zbl

[69] Michel Raynuad Flat Modules in Algebraic Geometry, Compos. Math., Volume 24 (1972) no. 1, pp. 11-31 | MR

[70] Carlos Simpson Subspaces of Moduli Spaces of Rank one Local Systems, Ann. Sci. Éc. Norm. Supér., Volume 26 (1993) no. 3, pp. 361-401 | DOI | MR | Numdam | Zbl

[71] Kenji Ueno Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lecture Notes in Mathematics, 439, Springer, 1975 | DOI

[72] Dror Varolin Three Variations on a Theme in Complex Analytic Geometry, Analytic And Algebraic Geometry: Common Problems, Different Methods (IAS/Park City Mathematics Series), Volume 17, American Mathematical Society; Institute for Advanced Studies (2010), pp. 183-295 | DOI | MR | Zbl

[73] Eckart Viehweg Rational Singularities of Higher Dimensional Schemes, Proc. Am. Math. Soc., Volume 63 (1977) no. 1, pp. 6-8 | DOI | MR | Zbl

[74] Eckart Viehweg Weak Positivity and the Additivity of the Kodaira Dimension for Certain Fibre Spaces, Algebraic Varieties and Analytic Varieties (Advanced Studies in Pure Mathematics), Volume 1, Mathematical Society of Japan (1983), pp. 329-353 | DOI | MR | Zbl

[75] Eckart Viehweg Quasi-projective Moduli for Polarized Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 30, Springer, 1995 | DOI

[76] Claire Voisin Théorie de Hodge et géométrie algébrique complexe, Contributions in Mathematical and Computational Sciences, 10, Société Mathématique de France; EDP Sciences, 2002

[77] Botong Wang Torsion Points on the Cohomology Jump Loci of Compact Kähler Manifolds, Math. Res. Lett., Volume 23 (2016) no. 2, pp. 545-563 | DOI | Zbl

[78] Xiangyu Zhou; Langfeng Zhu An optimal L2 Extension Theorem on Weakly Pseudoconvex Kähler Manifolds, J. Differ. Geom., Volume 110 (2018) no. 1, pp. 135-186 | Zbl

[79] Kang Zuo Kodaira Dimension and Chern Hyperbolicity of the Shafarevich Maps for Representations of $π_{1}$ of Compact Kähler Manifolds, J. Reine Angew. Math., Volume 472 (1996) no. 2, pp. 139-156 | Zbl

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