[Finitude des fibrations lagrangiennes à invariants fixes]
Nous démontrons la finitude des classes de déformation des fibrations lagrangiennes hyperkählériennes, de dimension quelconque, avec constante de Fujiki et discriminant du réseau de Beauville–Bogomolov–Fujiki fixes. Nous montrons également qu'il n'y a qu'un nombre fini de classes de déformation des fibrations lagrangiennes hyperkählériennes avec un fibré en droite ample de degré donné sur la fibre générale de la fibration.
We prove finiteness of the deformation classes of hyperkähler Lagrangian fibrations in any fixed dimension with fixed Fujiki constant and discriminant of the Beauville–Bogomolov–Fujiki lattice. We also prove there are only finitely many deformation classes of hyperkähler Lagrangian fibrations with an ample line bundle of a given degree on the general fibre of the fibration.
Accepté le :
Publié le :
@article{CRMATH_2016__354_7_707_0,
author = {Kamenova, Ljudmila},
title = {Finiteness of {Lagrangian} fibrations with fixed invariants},
journal = {Comptes Rendus. Math\'ematique},
pages = {707--711},
publisher = {Elsevier},
volume = {354},
number = {7},
year = {2016},
doi = {10.1016/j.crma.2015.12.019},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.12.019/}
}
TY - JOUR AU - Kamenova, Ljudmila TI - Finiteness of Lagrangian fibrations with fixed invariants JO - Comptes Rendus. Mathématique PY - 2016 SP - 707 EP - 711 VL - 354 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2015.12.019/ DO - 10.1016/j.crma.2015.12.019 LA - en ID - CRMATH_2016__354_7_707_0 ER -
%0 Journal Article %A Kamenova, Ljudmila %T Finiteness of Lagrangian fibrations with fixed invariants %J Comptes Rendus. Mathématique %D 2016 %P 707-711 %V 354 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2015.12.019/ %R 10.1016/j.crma.2015.12.019 %G en %F CRMATH_2016__354_7_707_0
Kamenova, Ljudmila. Finiteness of Lagrangian fibrations with fixed invariants. Comptes Rendus. Mathématique, Tome 354 (2016) no. 7, pp. 707-711. doi : 10.1016/j.crma.2015.12.019. http://www.numdam.org/articles/10.1016/j.crma.2015.12.019/
[1] Compact Complex Surfaces, EMG, vol. 3, Springer, Berlin, 2004
[2] Einstein Manifolds, Springer-Verlag, New York, 1987
[3] A superficial working guide to deformations and moduli (Farkas, G.; Morrison, I., eds.), Handbook of Moduli, Volume III, The Advanced Lectures in Mathematics Series, vol. 26, International Press, 2013, pp. 161-216 | arXiv
[4] Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for K3 surfaces, and the Tate conjecture, 2014 | arXiv
[5] Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom., Volume 3 (1994), pp. 295-345
[6] On the de Rham cohomology group of a compact Kähler symplectic manifold, Adv. Stud. Pure Math., Volume 10 (1987), pp. 105-165
[7] Intersection numbers of extremal rays on holomorphic symplectic varieties, Asian J. Math., Volume 14 (2010), pp. 303-322
[8] Compact hyper-Kähler manifolds: basic results, Invent. Math., Volume 135 (1999) no. 1, pp. 63-113 (Erratum: Invent. Math., 152, 1, 2003, pp. 209-212)
[9] Compact hyperkähler manifolds, Lectures from the Summer School held in Nordfjordeid, June 2001 (Universitext), Springer-Verlag, Berlin (2003), pp. 161-225
[10] Finiteness results for hyperkähler manifolds, J. Reine Angew. Math., Volume 558 (2003), pp. 15-22 | arXiv
[11] Families of Lagrangian fibrations on hyperkaehler manifolds, Adv. Math., Volume 260 (2014), pp. 401-413 | arXiv
[12] On fibre space structures of a projective irreducible symplectic manifold, Topology, Volume 38 (1999) no. 1, pp. 79-83 (also in Addendum Topology, 40, 2, 2001, pp. 431-432) | arXiv
[13] A finiteness theorem for Lagrangian fibrations | arXiv
[14] Cohomology of compact hyperkähler manifolds (87 p) | arXiv
Cité par Sources :
