Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Frédéric Campana; Thomas Peternell

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 1, page 41-74
  • ISSN: 0037-9484

Abstract

top
We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold X have a pseudo-effective (instead of generically nef) determinant. A first consequence is that X is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of X is not covered by compact positive-dimensional analytic subsets, then X is of general type if χ ( O X ) ≠ 0 . We finally show that if L is a numerically trivial line bundle on X , and if K X + L is ℚ -effective, then so is K X itself. The proof of this result rests on Simpson’s work on jumping loci of numerically trivial line bundles, and Viehweg’s cyclic covers. This last result is central, and has been recently extended, using the very same ingredients, to the case of log-canonical pairs.

How to cite

top

Campana, Frédéric, and Peternell, Thomas. "Geometric stability of the cotangent bundle and the universal cover of a projective manifold." Bulletin de la Société Mathématique de France 139.1 (2011): 41-74. <http://eudml.org/doc/272521>.

@article{Campana2011,
abstract = {We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold $X$ have a pseudo-effective (instead of generically nef) determinant. A first consequence is that $X$ is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of $X$ is not covered by compact positive-dimensional analytic subsets, then $X$ is of general type if $\chi (O_X)\ne 0$. We finally show that if $L$ is a numerically trivial line bundle on $X$, and if $K_X+L$ is $\mathbb \{Q\}$-effective, then so is $K_X$ itself. The proof of this result rests on Simpson’s work on jumping loci of numerically trivial line bundles, and Viehweg’s cyclic covers. This last result is central, and has been recently extended, using the very same ingredients, to the case of log-canonical pairs.},
author = {Campana, Frédéric, Peternell, Thomas},
journal = {Bulletin de la Société Mathématique de France},
keywords = {bundle; pseudo-effective line bundle; Moishezon-Iitaka-‘Kodaira’ dimension; universal cover; uniruledness},
language = {eng},
number = {1},
pages = {41-74},
publisher = {Société mathématique de France},
title = {Geometric stability of the cotangent bundle and the universal cover of a projective manifold},
url = {http://eudml.org/doc/272521},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Campana, Frédéric
AU - Peternell, Thomas
TI - Geometric stability of the cotangent bundle and the universal cover of a projective manifold
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 1
SP - 41
EP - 74
AB - We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold $X$ have a pseudo-effective (instead of generically nef) determinant. A first consequence is that $X$ is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of $X$ is not covered by compact positive-dimensional analytic subsets, then $X$ is of general type if $\chi (O_X)\ne 0$. We finally show that if $L$ is a numerically trivial line bundle on $X$, and if $K_X+L$ is $\mathbb {Q}$-effective, then so is $K_X$ itself. The proof of this result rests on Simpson’s work on jumping loci of numerically trivial line bundles, and Viehweg’s cyclic covers. This last result is central, and has been recently extended, using the very same ingredients, to the case of log-canonical pairs.
LA - eng
KW - bundle; pseudo-effective line bundle; Moishezon-Iitaka-‘Kodaira’ dimension; universal cover; uniruledness
UR - http://eudml.org/doc/272521
ER -

References

top
  1. [1] S. Boucksom – « Cônes positifs des variétés complexes compactes », thèse de doctorat, Université de Grenoble, 2002. 
  2. [2] S. Boucksom, J.-P. Demailly, M. Paun & T. Peternell – « The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension », to appear in J. Alg. Geom., 2011. Zbl1267.32017MR3019449
  3. [3] F. Campana – « Fundamental group and positivity of cotangent bundles of compact Kähler manifolds », J. Algebraic Geom.4 (1995), p. 487–502. Zbl0845.32027MR1325789
  4. [4] F. Campana, J. A. Chen & T. Peternell – « Strictly nef divisors », Math. Ann.342 (2008), p. 565–585. Zbl1154.14004MR2430991
  5. [5] F. Campana & Q. Zhang – « Compact Kähler threefolds of π 1 -general type », in Recent progress in arithmetic and algebraic geometry, Contemp. Math., vol. 386, Amer. Math. Soc., 2005, p. 1–12. Zbl1216.32011MR2182767
  6. [6] J.-P. Demailly – « Complex analytic and algebraic geometry », http://www-fourier.ujf-grenoble.fr/~demailly/books.html. 
  7. [7] J.-P. Demailly, T. Peternell & M. Schneider – « Holomorphic line bundles with partially vanishing cohomology », in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., 1996, p. 165–198. Zbl0859.14005MR1360502
  8. [8] H. Esnault – « Fibre de Milnor d’un cône sur une courbe plane singulière », Invent. Math.68 (1982), p. 477–496. Zbl0475.14018MR669426
  9. [9] D. Huybrechts & M. Lehn – The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, 1997. Zbl0872.14002MR1450870
  10. [10] Y. Kawamata – « Minimal models and the Kodaira dimension of algebraic fiber spaces », J. reine angew. Math. 363 (1985), p. 1–46. Zbl0589.14014MR814013
  11. [11] —, « Pluricanonical systems on minimal algebraic varieties », Invent. Math.79 (1985), p. 567–588. Zbl0593.14010MR782236
  12. [12] —, « Moderate degenerations of algebraic surfaces », in Complex algebraic varieties (Bayreuth, 1990), Lecture Notes in Math., vol. 1507, Springer, 1992, p. 113–132. Zbl0774.14032MR1178723
  13. [13] J. Kollár – « Shafarevich maps and plurigenera of algebraic varieties », Invent. Math.113 (1993), p. 177–215. Zbl0819.14006MR1223229
  14. [14] A. Langer – « Semistable sheaves in positive characteristic », Ann. of Math.159 (2004), p. 251–276. Zbl1080.14014MR2051393
  15. [15] J. Li & S.-T. Yau – « Hermitian-Yang-Mills connection on non-Kähler manifolds », in Mathematical aspects of string theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, 1987, p. 560–573. Zbl0664.53011MR915839
  16. [16] V. B. Mehta & A. Ramanathan – « Semistable sheaves on projective varieties and their restriction to curves », Math. Ann. 258 (1981/82), p. 213–224. Zbl0473.14001MR649194
  17. [17] Y. Miyaoka – « Deformations of a morphism along a foliation and applications », in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., 1987, p. 245–268. Zbl0659.14008MR927960
  18. [18] —, « Relative deformations of morphisms and applications to fibre spaces », Comment. Math. Univ. St. Paul.42 (1993), p. 1–7. Zbl0813.14001MR1223183
  19. [19] Y. Miyaoka & S. Mori – « A numerical criterion for uniruledness », Ann. of Math.124 (1986), p. 65–69. Zbl0606.14030MR847952
  20. [20] Y. Miyaoka & T. Peternell – Geometry of higher-dimensional algebraic varieties, DMV Seminar, vol. 26, Birkhäuser, 1997. Zbl0865.14018MR1468476
  21. [21] C. Mourougane – « Théorèmes d’annulation générique pour les fibrés vectoriels semi-négatifs », Bull. Soc. Math. France127 (1999), p. 115–133. Zbl0939.32020MR1700471
  22. [22] Y. Namikawa & J. H. M. Steenbrink – « Global smoothing of Calabi-Yau threefolds », Invent. Math.122 (1995), p. 403–419. Zbl0861.14036MR1358982
  23. [23] N. I. Shepherd-Barron – « Miyaoka’s theorem on the seminegativity of T X », Astérisque211 (1992), p. 103–114. Zbl0809.14034
  24. [24] —, « Semi-stability and reduction mod p », Topology37 (1998), p. 659–664. Zbl0926.14021MR1604907
  25. [25] C. Simpson – « Subspaces of moduli spaces of rank one local systems », Ann. Sci. École Norm. Sup.26 (1993), p. 361–401. Zbl0798.14005MR1222278
  26. [26] E. Viehweg – « Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces », in Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math., vol. 1, North-Holland, 1983, p. 329–353. Zbl0513.14019MR715656

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.