Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for -type and -type.
Nous décrivons quels fibrés en droites gros et nefs, sur des variétés symplectiques irréductibles, ont des diviseurs de base, sous certaines conditions relatives au type de déformation dont nous nous attendons à ce qu’elles soient vraies pour toutes les variétés symplectiques irréductibles. En particulier, nous montrons que de tels diviseurs de base sont toujours réduits et irréductibles. Nous appliquons ces résultats pour comprendre le comportement après déformation des diviseurs de base des fibrés en droites gros et nefs. Nous terminons en donnant une déscription très explicite pour les variétés de types et .
Revised:
Accepted:
Published online:
Keywords: Irreducible symplectic varieties, hyperkähler manifolds, base divisors, Fujita’s conjecture
Mot clés : Variétés symplectiques irréductibles, variétés hyperkähleriennes, diviseurs de base, conjecture de Fujita
Rieß, Ulrike 1
@article{AIF_2023__73_2_609_0, author = {Rie{\ss}, Ulrike}, title = {Base divisors of big and nef line bundles on irreducible symplectic varieties}, journal = {Annales de l'Institut Fourier}, pages = {609--633}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {73}, number = {2}, year = {2023}, doi = {10.5802/aif.3530}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3530/} }
TY - JOUR AU - Rieß, Ulrike TI - Base divisors of big and nef line bundles on irreducible symplectic varieties JO - Annales de l'Institut Fourier PY - 2023 SP - 609 EP - 633 VL - 73 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3530/ DO - 10.5802/aif.3530 LA - en ID - AIF_2023__73_2_609_0 ER -
%0 Journal Article %A Rieß, Ulrike %T Base divisors of big and nef line bundles on irreducible symplectic varieties %J Annales de l'Institut Fourier %D 2023 %P 609-633 %V 73 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3530/ %R 10.5802/aif.3530 %G en %F AIF_2023__73_2_609_0
Rieß, Ulrike. Base divisors of big and nef line bundles on irreducible symplectic varieties. Annales de l'Institut Fourier, Volume 73 (2023) no. 2, pp. 609-633. doi : 10.5802/aif.3530. https://aif.centre-mersenne.org/articles/10.5802/aif.3530/
[1] Rational curves on hyperkähler manifolds, Int. Math. Res. Not., Volume 2015 (2015) no. 23, pp. 13009-13045 | DOI
[2] Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkähler geometry (2016) (https://arxiv.org/abs/1604.03927)
[3] Effective freeness and point separation for adjoint bundles, Invent. Math., Volume 122 (1995) no. 2, pp. 291-308 | DOI | MR | Zbl
[4] Variétés kähleriennes dont la première classe de Chern est nulle, J. Differ. Geom., Volume 18 (1983), pp. 755-782 | Zbl
[5] Divisorial Zariski decompositions on compact complex manifolds., Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 1, pp. 45-76 | DOI | Numdam | MR | Zbl
[6] Remarks on Kawamata’s effective non-vanishing conjecture for manifolds with trivial first Chern classes (2016) (https://arxiv.org/abs/1612.00184v2)
[7] On the de Rham cohomology group of a compact Kähler symplectic manifold, Algebraic geometry (Sendai, 1985) (Advanced Studies in Pure Mathematics), Volume 10, Kinokuniya Company Ltd.; North-Holland, 1987 | DOI | Zbl
[8] Base manifolds for Lagrangian fibrations on hyperkähler manifolds., Int. Math. Res. Not., Volume 2014 (2014) no. 19, pp. 5483-5487 | DOI | Zbl
[9] Calabi–Yau Manifolds and Related Geometries. Lectures at a summer school in Nordfjordeid, Norway, June 2001., Universitext, Springer, 2003
[10] Birational symplectic manifolds and their deformations, J. Differ. Geom., Volume 45 (1997) no. 3, pp. 488-513 | MR | Zbl
[11] Compact hyperkähler manifolds: Basic results, Invent. Math., Volume 135 (1999) no. 1, pp. 63-113 | DOI | Zbl
[12] The Kähler cone of a compact hyperkähler manifold, Math. Ann., Volume 326 (2003) no. 3, pp. 499-513 | DOI | Zbl
[13] Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, 158, Cambridge University Press, 2016 | DOI
[14] Base manifolds for fibrations of projective irreducible symplectic manifolds, Invent. Math., Volume 174 (2008) no. 3, pp. 625-644 | DOI | MR | Zbl
[15] Pullbacks of hyperplane sections for Lagrangian fibrations are primitive (2016) (https://arxiv.org/abs/1612.07378v2)
[16] Positivity in Algebraic Geometry. II. Positivity for Vector Bundles, and Multiplier Ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 49, Springer, 2004 | DOI
[17] A survey of Torelli and monodromy results for holomorphic-symplectic varieties, Complex and differential geometry. Conference held at Leibniz Universität Hannover, Germany, September 14–18, 2009. Proceedings (Springer Proceedings in Mathematics), Volume 8, Springer, 2011, pp. 257-322 | MR | Zbl
[18] Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections, Kyoto J. Math., Volume 53 (2013) no. 2, pp. 345-403 | DOI | MR | Zbl
[19] A proof of the Kawamata-Morrison cone conjecture for holomorphic symplectic varieties of or generalized Kummer deformation type, Int. Math. Res. Not., Volume 2015 (2015) no. 24, pp. 13563-13574 | DOI | MR | Zbl
[20] On isotropic divisors on irreducible symplectic manifolds., Higher dimensional algebraic geometry. In honour of Professor Yujiro Kawamata’s sixtieth birthday. Proceedings of the conference, Tokyo, Japan, January 7–11, 2013, Mathematical Society of Japan, 2017, pp. 291-312 | DOI | MR | Zbl
[21] Families of K-3 surfaces, Nagoya Math. J., Volume 48 (1972), pp. 1-17 | DOI | MR | Zbl
[22] A note on the Kähler and Mori cones of hyperkähler manifolds, Asian J. Math., Volume 19 (2015) no. 4, pp. 583-592 | DOI | MR | Zbl
[23] Abelian varieties. With appendices by C. P. Ramanujam and Yuri Manin. Corrected reprint of the 2nd ed. 1974., American Mathematical Society; Tata Institute of Fundamental Research, 2008
[24] Desingularized moduli spaces of sheaves on a , J. Reine Angew. Math., Volume 512 (1999), pp. 49-117 | MR | Zbl
[25] A new six-dimensional irreducible symplectic variety, J. Algebr. Geom., Volume 12 (2003) no. 3, pp. 435-505 | MR | Zbl
[26] Compact hyperkähler manifolds: an introduction, available online, 2013
[27] On the non-divisorial base locus of big and nef line bundles of K3-type varieties (2020) (https://arxiv.org/abs/1901.08037, to appear in Proc. R. Soc. Edinb., Sect. A, Math.)
[28] Projective models of K-3 surfaces, Am. J. Math., Volume 96 (1974), pp. 602-639 | DOI | MR | Zbl
[29] Hyperkähler SYZ conjecture and semipositive line bundles, Geom. Funct. Anal., Volume 19 (2010) no. 5, pp. 1481-1493 | DOI | Zbl
Cited by Sources: