Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
EnglishFrançais
Compositio Mathematica Compositio Mathematica

Article contents

The Hodge diamond of O’Grady’s six-dimensional example

Part of: Surfaces and higher-dimensional varieties Birational geometry (Co)homology theory

Published online by Cambridge University Press:  21 March 2018

Giovanni Mongardi ,
Antonio Rapagnetta  and
Giulia Saccà
Giovanni Mongardi
Affiliation:
Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, Piazza di Porta San Donato 5, Bologna 40126, Italia email giovanni.mongardi2@unibo.it
Antonio Rapagnetta
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma 00133, Italia email rapagnet@axp.mat.uniroma2.it
Giulia Saccà
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, NY 11974-3651, USA email giulia.sacca@stonybrook.edu
Corresponding
E-mail address:
giovanni.mongardi2@unibo.it
rapagnet@axp.mat.uniroma2.it
giulia.sacca@stonybrook.edu
Get access
Rights & Permissions[Opens in a new window]

Abstract

We realize O’Grady’s six-dimensional example of an irreducible holomorphic symplectic (IHS) manifold as a quotient of an IHS manifold of $\text{K3}^{[3]}$ type by a birational involution, thereby computing its Hodge numbers.

Keywords

irreducible holomorphic symplectic manifolds Hodge numbers O’Grady’s six-dimensional manifold

MSC classification

Primary: 14J40: $n$-folds ($n>4$)
Secondary: 14E07: Birational automorphisms, Cremona group and generalizations 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 5 , May 2018 , pp. 984 - 1013
Copyright
© The Authors 2018 

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

References

Arbarello, E. and Saccà, G., Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties , Adv. Math. (2018), doi:10.1016/j.aim.2018.02.003.Google Scholar
Beauville, A., Variétés Kählériennes dont la première classe de Chern est nulle , J. Differential Geom. 18 (1983), 755–782.Google Scholar
Beauville, A., Counting rational curves on K3 surfaces , Duke Math. J. 97 (1999), 99–108.Google Scholar
Bogomolov, F., Hamiltonian Kähler manifolds , Soviet Math. Dokl. 19 (1978), 1462–1465.Google Scholar
Boissiére, S. and Sarti, A., A note on automorphisms and birational transformations of holomorphic symplectic manifolds , Proc. Amer. Math. Soc. 140 (2012), 4053–4062.Google Scholar
Collingwood, D. H. and McGovern, W. M., Nilpotent orbits in semisimple Lie algebra, Van Nostrand Reinhold Mathematics Series (Van Nostrand Reinhold, New York, 1993).Google Scholar
Fujiki, A. and Nakano, S., Supplement to ‘On the inverse of monoidal transformation’ , Publ. Res. Inst. Math. Sci. 7 (1971/72), 637–644.Google Scholar
Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, second edition (Springer, Berlin, 1998).Google Scholar
Göttsche, L., The Betti numbers of the Hilbert scheme of points on a smooth projective surface , Math. Ann. 286 (1990), 193–207.CrossRefGoogle Scholar
Göttsche, L. and Sörgel, W., Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces , Math. Ann. 296 (1993), 235–245.Google Scholar
Greb, D., Lehn, C. and Rollenske, S., Lagrangian fibrations of hyperkähler manifolds, on a question of Beauville , Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), 375–403.Google Scholar
Grothendieck, A., Revètements étales et groupe fondamental , in Fasc. I: Exposès 1 à 5. Séminaire de Géométrie Algébrique, 1960/61, Troisième édition (corrigée; Institut des Hautes Études Scientifiques, Paris, 1963).Google Scholar
Hesselink, W., The normality of closures of orbits in a Lie algebra , Comment. Math. Helv. 54 (1979), 105–110.CrossRefGoogle Scholar
Höhn, G. and Mason, G., Finite groups of symplectic automorphisms of hyperkähler manifolds of type $K3^{[2]}$ , Preprint (2014), arXiv:1409.6055.Google Scholar
Huybrechts, D., Compact hyperkähler manifolds: basic results , Invent. Math. 135 (1999), 63–113.CrossRefGoogle Scholar
Huybrechts, D., The Kähler cone of a compact hyperkähler manifold , Math. Ann. 326 (2003), 499–513.Google Scholar
Huybrechts, D. and Lehn, M., The geometry of moduli spaces of sheaves (Cambridge University Press, Cambridge, 2010).CrossRefGoogle Scholar
Kawatani, K., On the birational geometry for irreducible symplectic 4-folds related to the Fano schemes of lines, Preprint (2009), arXiv:0906.0654.Google Scholar
Kollar, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134 (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
Lehn, M. and Sorger, C., La singularité de O’Grady , J. Algebraic Geom. 15 (2006), 753–770.CrossRefGoogle Scholar
Le Potier, J., Systèmes cohérents et structures de niveau , Astérisque 214 (1993).Google Scholar
Markman, E., Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces , J. Reine Angew. Math. 544 (2002), 61–82.Google Scholar
Menet, G., Beauville–Bogomolov lattice for a singular symplectic variety of dimension 4 , J. Pure Appl. Algebra 219 (2015), 1455–1495.CrossRefGoogle Scholar
Mongardi, G., Towards a classification of symplectic automorphisms on manifolds of K3[n] type , Math. Z. 282 (2016), 651–662.Google Scholar
Mongardi, G. and Wandel, M., Automorphisms of O’Grady’s manifolds acting trivially on cohomology , Algebr. Geom. 4 (2017), 104–119.Google Scholar
Mozgovyy, S., The Euler number of O’Grady’s ten dimensional symplectic manifold, PhD Thesis, Mainz Universität (2006).Google Scholar
Mukai, S., Moduli of vector bundles on K3 surfaces and symplectic manifolds , Sugaku Expositions 1 (1988), 139–174.Google Scholar
Nakano, S., On the inverse of monoidal transformations , Publ. Res. Inst. Math. Sci. 6 (1971), 483–502.CrossRefGoogle Scholar
O’Grady, K. G., Desingularized moduli spaces of sheaves on a K3 , J. Reine Angew. Math. 512 (1999), 49–117.Google Scholar
O’Grady, K. G., A new six-dimensional irreducible symplectic variety , J. Algebraic Geom. 12 (2003), 435–505.CrossRefGoogle Scholar
O’Grady, K. G., Higher dimensional analogues of K3 surfaces. Current developments in algebraic geometry , Publ. Res. Inst. Math. Sci. 59 (2012), 257–293.Google Scholar
Perego, A. and Rapagnetta, A., Deformation of the O’Grady moduli spaces , J. Reine Angew. Math. 678 (2013), 1–34.CrossRefGoogle Scholar
Perego, A. and Rapagnetta, A., Factoriality properties of moduli spaces of sheaves on abelian and K3 surfaces , Int. Math. Res. Not. IMRN 2014 (2014), 643–680.Google Scholar
Rapagnetta, A., Topological invariants of O’Grady’s six dimensional irreducible symplectic variety, PhD thesis, Università degli studi di Roma Tor Vergata, 2004.Google Scholar
Rapagnetta, A., On the Beauville form of known irreducible symplectic varieties , Math. Ann. 340 (2008), 77–95.CrossRefGoogle Scholar
Salamon, S., On the cohomology of Kähler and hyperkähler manifolds , Topology 35 (1996), 137–155.CrossRefGoogle Scholar
Sawon, J., Rozansky–Witten invariants of hyperkähler manifolds, PhD thesis, University of Cambridge, 1999.Google Scholar
Voisin, C., Hodge theory and complex algebraic geometry. I. Translated from the French by Leila Schneps, Cambridge Studies in Advanced Mathematics, vol. 76; reprint of the 2002 English edition (Cambridge University Press, Cambridge, 2007).Google Scholar
Yoshioka, K., Moduli of vector bundles on algebraic surfaces , Sugaku Expositions 20 (2007), 111–135.Google Scholar

Altmetric attention score

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 8
Total number of PDF views: 125 *
View data table for this chart

* Views captured on Cambridge Core between 21st March 2018 - 17th January 2021. This data will be updated every 24 hours.

2 Cited by
Hostname: page-component-77fc7d77f9-n279q Total loading time: 0.849 Render date: 2021-01-17T01:00:08.236Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags last update: Sun Jan 17 2021 00:53:43 GMT+0000 (Coordinated Universal Time) Feature Flags: { "metrics": true, "metricsAbstractViews": false, "peerReview": true, "crossMark": true, "comments": true, "relatedCommentaries": true, "subject": true, "clr": true, "languageSwitch": true, "figures": false, "newCiteModal": false, "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

The Hodge diamond of O’Grady’s six-dimensional example
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

The Hodge diamond of O’Grady’s six-dimensional example
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

The Hodge diamond of O’Grady’s six-dimensional example
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *