Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-12T21:15:13.529Z Has data issue: false hasContentIssue false
  • English
  • Français

Formality conjecture for K3 surfaces

Part of: Homological methods Families, fibrations Local theory Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  23 April 2019

Nero Budur  and
Ziyu Zhang
Nero Budur
Affiliation:
KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium email [email protected] https://www.kuleuven.be/wis/algebra/budur
Ziyu Zhang
Affiliation:
Institut für algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany email [email protected] https://ziyuzhang.github.io
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra $\operatorname{RHom}^{\bullet }(F,F)$ is formal for any sheaf $F$ polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

Keywords

K3 surfacesstable sheavesformalityDG algebrasDG enhancements

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles
Secondary: 16E45: Differential graded algebras and applications 14B05: Singularities 14J28: $K3$ surfaces and Enriques surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 5 , May 2019 , pp. 902 - 911
Copyright
© The Authors 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arbarello, E. and Saccà, G., Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties , Adv. Math. 329 (2018), 649–703, doi:10.1016/j.aim.2018.02.003.Google Scholar
Bayer, A. and Macrì, E., Projectivity and birational geometry of Bridgeland moduli spaces , J. Amer. Math. Soc. 27 (2014), 707–752, doi:10.1090/S0894-0347-2014-00790-6.Google Scholar
Bayer, A. and Macrì, E., MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations , Invent. Math. 198 (2014), 505–590,doi:10.1007/s00222-014-0501-8.Google Scholar
Bellamy, G. and Schedler, T., Symplectic resolutions of quiver varieties and character varieties, Preprint (2016), arXiv:1602.00164 [math.AG].Google Scholar
Bondal, A. and Kapranov, M., Enhanced triangulated categories , Math. USSR-Sb. 70 (1991), 93–107, doi:10.1070/SM1991v070n01ABEH001253.Google Scholar
Bridgeland, T., Stability conditions on K3 surfaces , Duke Math. J. 141 (2008), 241–291,doi:10.1215/S0012-7094-08-14122-5.Google Scholar
Budur, N., Rational singularities, quiver moment maps, and representations of surface groups, Preprint (2018), arXiv:1809.05180 [math.AG].Google Scholar
Budur, N. and Zhang, Z., Formality conjecture for K3 surfaces, Preprint (2018), arXiv:1803.03974v3 [math.AG].Google Scholar
Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D., Real homotopy theory of Kähler manifolds , Invent. Math. 29 (1975), 1073–1089, doi:10.1007/BF01389853.Google Scholar
Goldman, W. and Millson, J., The deformation theory of representations of fundamental groups of compact Kähler manifolds , Bull. Amer. Math. Soc. (N.S.) 18 (1988), 153–158,doi:10.1090/S0273-0979-1988-15631-5.Google Scholar
Hovey, M., Model categories, Mathematical Surveys and Monographs, vol. 63 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Huybrechts, D., Fourier-Mukai transforms in algebraic geometry (Clarendon Press, Oxford, 2006).Google Scholar
Huybrechts, D. and Lehn, M., The geometry of moduli spaces of sheaves (Cambridge University Press, Cambridge, 2010).Google Scholar
Huybrechts, D., Macrì, E. and Stellari, P., Stability conditions for generic K3 categories , Compos. Math. 144 (2008), 134–162, doi:10.1112/S0010437X07003065.Google Scholar
Kaledin, D. and Lehn, M., Local structure of hyperkähler singularities in O’Grady’s examples , Mosc. Math. J. 7 (2007), 653–672; 766–767, http://www.ams.org/distribution/mmj/vol7-4-2007/kaledin-lehn.pdf.Google Scholar
Kaledin, D., Lehn, M. and Sorger, C., Singular symplectic moduli spaces , Invent. Math. 164 (2006), 591–614, doi:10.1007/s00222-005-0484-6.Google Scholar
Kontsevich, M., Deformation quantization of Poisson manifolds , Lett. Math. Phys. 66 (2003), 157–216, doi:10.1023/B:MATH.0000027508.00421.bf.Google Scholar
Lekili, Y. and Ueda, K., Homological mirror symmetry for K3 surfaces via moduli of $A_{\infty }$ -structures, Preprint (2018), arXiv:1806.04345 [math.AG].Google Scholar
Lunts, V. and Orlov, D., Uniqueness of enhancement for triangulated categories , J. Amer. Math. Soc. 23 (2010), 853–908, doi:10.1090/S0894-0347-10-00664-8.Google Scholar
Meachan, C. and Zhang, Z., Birational geometry of singular moduli spaces of O’Grady type , Adv. Math. 296 (2016), 210–267, doi:10.1016/j.aim.2016.02.036.Google Scholar
Seidel, P., Homological mirror symmetry for the quartic surface , Mem. Amer. Math. Soc. 236 (2015), doi:10.1090/memo/1116.Google Scholar
Tabuada, G., Une structure de cataégorie de modèles de Quillen sur la catégorie des dg-catégories , C. R. Math. Acad. Sci. Paris 340 (2005), 15–19, doi:10.1016/j.crma.2004.11.007.Google Scholar
Toda, Y., Non-commutative thickening of moduli spaces of stable sheaves , Compos. Math. 153 (2017), 1153–1195, doi:10.1112/S0010437X17007047.Google Scholar
Toda, Y., Moduli stacks of semistable sheaves and representations of Ext-quivers , Geom. Topol. 22 (2018), 3083–3144, doi:10.2140/gt.2018.22.3083.Google Scholar
Toën, B., Lectures on dg-categories , in Topics in algebraic and topological K-theory, Lecture Notes in Mathematics, vol. 2008 (Springer, Berlin, 2011), 243–302,doi: 10.1007/978-3-642-15708-0_5.Google Scholar
Toën, B., Problèmes de modules formels , in Séminaire Bourbaki. vol. 2015/2016, Astérisque vol. 390 (Société Mathématique de France, Paris, 2017), 1104–1119; Exposés. Exp. No. 1111, 199–244.Google Scholar
Yoshioka, K., Moduli spaces of stable sheaves on abelian surfaces , Math. Ann. 321 (2001), 817–884, doi:10.1007/s002080100255.Google Scholar
Yoshioka, K., Stability and the Fourier-Mukai transform. II , Compos. Math. 145 (2009), 112–142, doi:10.1112/S0010437X08003758.Google Scholar
Yoshioka, K., Fourier–Mukai duality for K3 surfaces via Bridgeland stability condition , J. Geom. Phys. 122 (2017), 103–118, doi:10.1016/j.geomphys.2016.08.011.Google Scholar
Zhang, Z., A note on formality and singularities of moduli spaces , Mosc. Math. J. 12 (2012), 863–879; 885, http://www.mathjournals.org/mmj/2012-012-004/2012-012-004-011.pdf.Google Scholar