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.

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 K3 [n] -type and Kum n -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 K3 [n] et Kum n .

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3530
Classification: 14M99
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

1 ETH Zürich Institute for Theoretical Studies Clausisusstrasse 47 8092 Zürich (Switzerland)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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] Amerik, Ekaterina; Verbitsky, Misha Rational curves on hyperkähler manifolds, Int. Math. Res. Not., Volume 2015 (2015) no. 23, pp. 13009-13045 | DOI

[2] Amerik, Ekaterina; Verbitsky, Misha Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkähler geometry (2016) (https://arxiv.org/abs/1604.03927)

[3] Angehrn, Urban; Siu, Yum-Tong Effective freeness and point separation for adjoint bundles, Invent. Math., Volume 122 (1995) no. 2, pp. 291-308 | DOI | MR | Zbl

[4] Beauville, Arnaud Variétés kähleriennes dont la première classe de Chern est nulle, J. Differ. Geom., Volume 18 (1983), pp. 755-782 | Zbl

[5] Boucksom, Sébastien 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] Cao, Yalong; Jiang, Chen Remarks on Kawamata’s effective non-vanishing conjecture for manifolds with trivial first Chern classes (2016) (https://arxiv.org/abs/1612.00184v2)

[7] Fujiki, Akira 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] Greb, Daniel; Lehn, Christian Base manifolds for Lagrangian fibrations on hyperkähler manifolds., Int. Math. Res. Not., Volume 2014 (2014) no. 19, pp. 5483-5487 | DOI | Zbl

[9] Gross, Mark; Huybrechts, Daniel; Joyce, Dominic Calabi–Yau Manifolds and Related Geometries. Lectures at a summer school in Nordfjordeid, Norway, June 2001., Universitext, Springer, 2003

[10] Huybrechts, Daniel Birational symplectic manifolds and their deformations, J. Differ. Geom., Volume 45 (1997) no. 3, pp. 488-513 | MR | Zbl

[11] Huybrechts, Daniel Compact hyperkähler manifolds: Basic results, Invent. Math., Volume 135 (1999) no. 1, pp. 63-113 | DOI | Zbl

[12] Huybrechts, Daniel The Kähler cone of a compact hyperkähler manifold, Math. Ann., Volume 326 (2003) no. 3, pp. 499-513 | DOI | Zbl

[13] Huybrechts, Daniel Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, 158, Cambridge University Press, 2016 | DOI

[14] Hwang, Jun-Muk Base manifolds for fibrations of projective irreducible symplectic manifolds, Invent. Math., Volume 174 (2008) no. 3, pp. 625-644 | DOI | MR | Zbl

[15] Kamenova, Ljudmila; Verbitsky, Misha Pullbacks of hyperplane sections for Lagrangian fibrations are primitive (2016) (https://arxiv.org/abs/1612.07378v2)

[16] Lazarsfeld, Robert 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] Markman, Eyal 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] Markman, Eyal 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] Markman, Eyal; Yoshioka, Kota A proof of the Kawamata-Morrison cone conjecture for holomorphic symplectic varieties of K3 [n] or generalized Kummer deformation type, Int. Math. Res. Not., Volume 2015 (2015) no. 24, pp. 13563-13574 | DOI | MR | Zbl

[20] Matsushita, Daisuke 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] Mayer, Alan L. Families of K-3 surfaces, Nagoya Math. J., Volume 48 (1972), pp. 1-17 | DOI | MR | Zbl

[22] Mongardi, Giovanni 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] Mumford, David 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] O’Grady, Kieran G. Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math., Volume 512 (1999), pp. 49-117 | MR | Zbl

[25] O’Grady, Kieran G. A new six-dimensional irreducible symplectic variety, J. Algebr. Geom., Volume 12 (2003) no. 3, pp. 435-505 | MR | Zbl

[26] O’Grady, Kieran G. Compact hyperkähler manifolds: an introduction, available online, 2013

[27] Rieß, Ulrike On the non-divisorial base locus of big and nef line bundles of K3 [2] -type varieties (2020) (https://arxiv.org/abs/1901.08037, to appear in Proc. R. Soc. Edinb., Sect. A, Math.)

[28] Saint-Donat, Bernard Projective models of K-3 surfaces, Am. J. Math., Volume 96 (1974), pp. 602-639 | DOI | MR | Zbl

[29] Verbitsky, Misha Hyperkähler SYZ conjecture and semipositive line bundles, Geom. Funct. Anal., Volume 19 (2010) no. 5, pp. 1481-1493 | DOI | Zbl

Cited by Sources: