Tohoku Mathematical Journal

On nef and semistable hermitian lattices, and their behaviour under tensor product

Yves André

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We study the behaviour of semistability under tensor product in various settings: vector bundles, euclidean and hermitian lattices (alias Humbert forms or Arakelov bundles), multifiltered vector spaces.

One approach to show that semistable vector bundles in characteristic zero are preserved by tensor product is based on the notion of nef vector bundles. We revisit this approach and show how far it can be transferred to hermitian lattices. J.-B. Bost conjectured that semistable hermitian lattices are preserved by tensor product. Using properties of nef hermitian lattices, we establish an inequality in that direction.

We axiomatize our method in the general context of monoidal categories, and then give an elementary proof of the fact that semistable multifiltered vector spaces (which play a role in diophantine approximation) are preserved by tensor product.

Article information

Tohoku Math. J. (2), Volume 63, Number 4 (2011), 629-649.

First available in Project Euclid: 6 January 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E39: Bilinear and Hermitian forms
Secondary: 14G25: Global ground fields 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05]


André, Yves. On nef and semistable hermitian lattices, and their behaviour under tensor product. Tohoku Math. J. (2) 63 (2011), no. 4, 629--649. doi:10.2748/tmj/1325886284.

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  • Y. André, Slope filtrations, Confluentes Math. 1 (2009), 1–85.
  • C. M. Barton, Tensor products of ample vector bundles in characteristic $p$, Amer. J. Math. 93 (1971), 429–438.
  • T. Borek, Successive minima and slopes of hermitian vector bundles over number fields, J. Number Theory 113 (2005), 380–388.
  • J.-B. Bost and K. Künnemann, Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers, Adv. Math. 223 (2010), 987–1106.
  • J.-B. Bost, H. Gillet and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), 903–1027.
  • H. Brenner, Slopes of vector bundles on projective curves and applications to tight closure problems, Trans. Amer. Math. Soc. 356 (2003), 371–392.
  • U. Bruzzo and D. Hernández Ruiprez, Semistability vs. nefness for (Higgs) vector bundles, Differential Geom. Appl. 24 (2006), 403–416.
  • B. Casselman, Stability of lattices and the partition of arithmetic quotients, Asian J. Math. 8 (2004), 607–637.
  • H. Chen, Maximal slope of tensor product of Hermitian vector bundles, J. Algebraic Geom. 18 (2009), 575–603.
  • R. Coulangeon, Tensor products of Hermitian lattices, Acta Arith. 92 (2000), 115–130.
  • R. Coulangeon, Voronoi theory over algebraic number fields, Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math. 37 (2001), 147–162.
  • E. De Shalit and O. Parzanchevski, On tensor products of semistable lattices, preprint Univ. Jerusalem 2006 (available at$\sim$deshalit/).
  • G. Faltings, Mumford-Stabilität in der algebraischen Geometrie, Proc. Intern. Congress Math. (Zürich 1994), 648–655, Birkhäuser, 1995.
  • G. Faltings and G. Wüstholz, Diophantine approximations on projective spaces, Invent. Math. 116 (1994), 109–138.
  • É. Gaudron, Pentes des fibrés vectoriels adéliques sur un corps global, Rend. Semin. Mat. Univ. Padova 119 (2008), 21–95.
  • É. Gaudron, Géométrie des nombres adélique et formes linéaires de logarithmes dans un groupe algébrique commutatif, Mémoire d'habilitation 2009 (unpublished, available at$\sim$gaudron/).
  • R. Grayson, Reduction theory using semistability, Comment. Math. Helv. 59 (1984), 600–634.
  • D. Hoffmann, On positive definite Hermitian forms, Manuscripta Math. 71 (1991), 399–429.
  • N. Hoffmann, Stability of Arakelov bundles and tensor products without global sections, Doc. Math. 8 (2003), 115–123.
  • N. Hoffmann, J. Jahnel and U. Stuhler, Generalized vector bundles on curves, J. Reine Angew. Math. 495 (1998), 35–60.
  • P. Humbert, Théorie de la réduction des formes quadratiques définies positives dans un corps algébrique $K$ fini, Comment. Math. Helv. 12 (1940), 263–306.
  • M. Icaza, Hermite constant and extreme forms for algebraic number fields, J. London Math. Soc. (2) 55 (1997), 11–22.
  • Y. Kim, On semistability of root lattices and perfect lattices, preprint Univ. Illinois 2009 (available at$\sim$ykim33/).
  • Y. Kitaoka, Arithmetic of quadratic forms, Cambridge Tracts in Math. 106, Cambridge Univ. Press, 1993.
  • S. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. 84 (1966), 293–344.
  • R. Lazarsfeld, Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals, Ergeb. der Math. und ihrer Grenzgebiete 49, Springer-Verlage, Berlin, 2004.
  • J. Martinet, Perfect lattices in Euclidean spaces, Grundlehren Math. Wiss. 327, Springer-Verlage, Berlin, 2003.
  • M. Maruyama, The theorem of Grauert-Mühlich-Spindler, Math. Ann. (1981), 317–333.
  • Y. Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai, 1985, 449–476, Adv. Stud. Pure Math. 10, North-Holland, 1987.
  • A. Moriwaki, Relative Bogomolov's inequality and the cone of positive divisors on the moduli space of stable curves, J. Amer. Math. Soc. 11 (1998), 569–600.
  • A. Moriwaki, Arithmetic height functions over finitely generated fields, Invent. Math. 140 (2000), 101–142.
  • M. Narasimhan and C. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540–567.
  • A. Pekker, On successive minima and the absolute Siegel's Lemma, J. Number Theory 128 (2007), 564–575.
  • S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. J. 36 (1984), 269–291.
  • M. Rapoport, Analogien zwischen den Modulräumen von Vektorbündeln und von Flaggen (DMV Tagung, Ulm, 1995), Jahresber. Deutsch. Math.-Verein. 99 (1997), 164–180.
  • D. Roy and J. Thunder, An absolute Siegel lemma, J. Reine Angew. Math. 476 (1996), 1–26, addendum et erratum, ibid. 508 (1999), 47–51.
  • C. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95.
  • U. Stuhler, Eine Bemerkung zur Reduktionstheorie quadratischen Formen, Arch. Math. 27 (1976), 604–610.
  • B. Totaro, Tensor products of semistables are semistable, Geometry and analysis on complex manifolds 242–250, World Sci. Publ., Singapore, 1994.
  • S. Zhang, Positive line bundles on arithmetic varieties J. Amer. Math. Soc. 8 (1995), 187–221.