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
Forum of Mathematics, Pi Forum of Mathematics, Pi

Article contents

THE KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES

Part of: Projective and enumerative geometry

Published online by Cambridge University Press:  06 June 2016

R. PANDHARIPANDE  and
R. P. THOMAS
R. PANDHARIPANDE
Affiliation:
Departement Mathematik, ETH Zürich, Switzerland; rahul@math.ethz.ch
R. P. THOMAS
Affiliation:
Department of Mathematics, Imperial College, UK; richard.thomas@imperial.ac.uk
Corresponding
E-mail address:
rahul@math.ethz.ch
richard.thomas@imperial.ac.uk
Save pdf (0.97 mb)

Abstract

We prove the KKV conjecture expressing Gromov–Witten invariants of $K3$ surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov–Witten/Pairs correspondence for $K3$ -fibered hypersurfaces of dimension 3 to reduce the KKV conjecture to statements about stable pairs on (thickenings of) $K3$ surfaces. Using degeneration arguments and new multiple cover results for stable pairs, we reduce the KKV conjecture further to the known primitive cases. Our results yield a new proof of the full Yau–Zaslow formula, establish new Gromov–Witten multiple cover formulas, and express the fiberwise Gromov–Witten partition functions of $K3$ -fibered 3-folds in terms of explicit modular forms.

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 4 , 2016 , e4
Check for updates
Creative Commons
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

References

Altman, A., Iarrobino, A. and Kleiman, S., ‘Irreducibility of the compactified Jacobian’, inReal and Complex Singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) (1977), 1–12.Google Scholar
Behrend, K., ‘Gromov–Witten invariants in algebraic geometry’, Invent. Math. 127 (1997), 601–617.CrossRefGoogle Scholar
Behrend, K. and Fantechi, B., ‘The intrinsic normal cone’, Invent. Math. 128 (1997), 45–88.CrossRefGoogle Scholar
Beauville, A., ‘Counting rational curves on K3 surfaces’, Duke Math. J. 97 (1999), 99–108.CrossRefGoogle Scholar
Borcherds, R., ‘The Gross–Kohnen–Zagier theorem in higher dimensions’, Duke Math. J. 97 (1999), 219–233.CrossRefGoogle Scholar
Bridgeland, T., ‘Hall algebras and curve-counting invariants’, J. Amer. Math. Soc. 24 (2011), 969–998.CrossRefGoogle Scholar
Bryan, J., Katz, S. and Leung, C., ‘Multiple covers and the integrality conjecture for rational curves in Calabi–Yau threefolds’, J. Algebraic Geom. 10 (2001), 549–568.Google Scholar
Bryan, J. and Leung, C., ‘The enumerative geometry of K3 surfaces and modular forms’, J. Amer. Math. Soc. 13 (2000), 371–410.CrossRefGoogle Scholar
Buchweitz, R.-O. and Flenner, H., ‘A semiregularity map for modules and applications to deformations’, Compos. Math. 137 (2003), 135–210.CrossRefGoogle Scholar
Candelas, P., de la Ossa, X., Green, P. and Parkes, L., ‘A pair of Calabi–Yau manifolds as an exactly soluble superconformal field theory’, Nuclear Phys. B 359 (1991), 21–74.CrossRefGoogle Scholar
Chen, X., ‘Rational curves on K3 surfaces’, J. Algebraic Geom. 8 (1999), 245–278.Google Scholar
Dolgachev, I. and Kondo, S., ‘Moduli of $K3$ surfaces and complex ball quotients, Lectures in Istambul’, Preprint, 2005, arXiv:math.AG/0511051.Google Scholar
Faber, C. and Pandharipande, R., ‘Hodge integrals and Gromov–Witten theory’, Invent. Math. 139 (2000), 173–199.CrossRefGoogle Scholar
Fantechi, B., Göttsche, L. and van Straten, D., ‘Euler number of the compactified Jacobian and multiplicity of rational curves’, J. Algebraic Geom. 8 (1999), 115–133.Google Scholar
Fulton, W., Intersection Theory (Springer, Berlin, 1998).CrossRefGoogle Scholar
Gopakumar, R. and Vafa, C., ‘ $M$ -theory and topological strings I’, Preprint, 1998, arXiv:hep-th/9809187.Google Scholar
Gopakumar, R. and Vafa, C., ‘ $M$ -theory and topological strings II’, Preprint, 1998, arXiv:hep-th/9812127.Google Scholar
Graber, T. and Pandharipande, R., ‘Localization of virtual classes’, Invent. Math. 135 (1999), 487–518.CrossRefGoogle Scholar
Huybrechts, D., Lectures on K3 Surfaces (Cambridge University Press, Cambridge, 2016).CrossRefGoogle Scholar
Huybrechts, D. and Thomas, R. P., ‘Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes’, Math. Ann. 346 (2010), 545–569.CrossRefGoogle Scholar
Illusie, L., Complexe cotangent et déformations I, Lecture Notes in Mathematics, 239 (Springer, Berlin, 1971).CrossRefGoogle Scholar
Joyce, D. and Song, Y., A Theory of Generalized Donaldson–Thomas Invariants, Memoirs of the American Mathematical Society, 217 (American Mathematical Society, 2012), iv+199 pp.Google Scholar
Katz, S., Klemm, A. and Pandharipande, R., ‘On the motivic stable pairs invariants of K3 surfaces, with an appendix by R. Thomas’, inK3 Surfaces and their Moduli (eds. Faber, C., Farkas, G. and van der Geer, G.) Birkhauser Progress in Mathematics, 315 (Birkhäuser, Basel, 2016), 111–146.CrossRefGoogle Scholar
Katz, S., Klemm, A. and Vafa, C., ‘ M-theory, topological strings, and spinning black holes’, Adv. Theor. Math. Phys. 3 (1999), 1445–1537.CrossRefGoogle Scholar
Kawai, T. and Yoshioka, K., ‘String partition functions and infinite products’, Adv. Theor. Math. Phys. 4 (2000), 397–485.CrossRefGoogle Scholar
Kiem, Y. H. and Li, J., ‘Localized virtual cycles by cosections’, J. Amer. Math. Soc. 26 (2013), 1025–1050.CrossRefGoogle Scholar
Klemm, A., Kreuzer, M., Riegler, E. and Scheidegger, E., ‘Topological string amplitudes, complete intersections Calabi–Yau spaces, and threshold corrections’, J. High Energy Phys. 05 023 (2005).Google Scholar
Klemm, A., Maulik, D., Pandharipande, R. and Scheidegger, E., ‘Noether–Lefschetz theory and the Yau–Zaslow conjecture’, J. Amer. Math. Soc. 23 (2010), 1013–1040.CrossRefGoogle Scholar
Kool, M. and Thomas, R. P., ‘Reduced classes and curve counting on surfaces I: theory’, Algebr. Geom. 1 (2014), 334–383.CrossRefGoogle Scholar
Kool, M. and Thomas, R. P., ‘Reduced classes and curve counting on surfaces II: calculations’, Algebr. Geom. 1 (2014), 384–399.CrossRefGoogle Scholar
Kudla, S. and Millson, J., ‘Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables’, Pub. IHES 71 (1990), 121–172.CrossRefGoogle Scholar
Lee, J. and Leung, C., ‘Yau–Zaslow formula for non-primitive classes in K3 surfaces’, Geom. Topol. 9 (2005), 1977–2012.CrossRefGoogle Scholar
Li, J., ‘Stable morphisms to singular schemes and relative stable morphisms’, J. Differential Geom. 57 (2001), 509–578.CrossRefGoogle Scholar
Li, J., ‘A degeneration formula for Gromov–Witten invariants’, J. Differential Geom. 60 (2002), 199–293.CrossRefGoogle Scholar
Li, J. and Wu, B., ‘Good degeneration of Quot-schemes and coherent systems’, Preprint, 2011, arXiv:1110.0390.Google Scholar
Maulik, D. and Pandharipande, R., ‘Gromov–Witten theory and Noether–Lefschetz theory’, inA Celebration of Algebraic Geometry, Clay Mathematics Proceedings, 18 (AMS, Providence, RI, 2010), 469–507.Google Scholar
Maulik, D., Pandharipande, R. and Thomas, R., ‘Curves on K3 surfaces and modular forms’, J. Topol. 3 (2010), 937–996.CrossRefGoogle Scholar
Oberdieck, G. and Pandharipande, R., ‘Curve counting on K3 × E, the Igusa cusp form 𝜒10 , and descendent integration’, in K3 Surfaces and their Moduli (eds. Faber, C., Farkas, G. and van der Geer, G.) Birkhauser Progress in Mathematics, 315 (Birkhäuser, Basel, 2016), 245–278.CrossRefGoogle Scholar
Pandharipande, R. and Pixton, A., ‘Gromov–Witten/Pairs correspondence for the quintic 3-fold’, J. Amer. Math. Soc. (2016), doi:10.1090/jams/858.CrossRefGoogle Scholar
Pandharipande, R. and Thomas, R. P., ‘Curve counting via stable pairs in the derived category’, Invent. Math. 178 (2009), 407–447.CrossRefGoogle Scholar
Pandharipande, R. and Thomas, R. P., ‘Stable pairs and BPS invariants’, J. Amer. Math. Soc. 23 (2010), 267–297.CrossRefGoogle Scholar
Pridham, J., ‘Semiregularity as a consequence of Goodwillie’s theorem’, Preprint, 2012, arXiv:1208.3111.Google Scholar
Schürg, T., Toën, B. and Vezzosi, G., ‘Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes’, Preprint, 2011, arXiv:1102.1150.Google Scholar
Siebert, B., ‘An update on (small) quantum cohomology’, inMirror Symmetry, III (Montreal, PQ, 1995), AMS/IP Studies in Advanced Mathematics, 10 (American Mathematical Society, Providence, RI, 1999), 279–312.Google Scholar
Toda, Y., ‘Curve counting theories via stable objects I. DT/PT correspondence’, J. Amer. Math. Soc. 23 (2010), 1119–1157.CrossRefGoogle Scholar
Toda, Y., ‘Stable pairs on local K3 surfaces’, J. Differential Geom. 92 (2012), 285–370.CrossRefGoogle Scholar
Wall, C. T. C., ‘On the orthogonal groups of unimodular quadratic forms’, Math. Ann. 147 (1962), 328–338.CrossRefGoogle Scholar
Yau, S.-T. and Zaslow, E., ‘BPS states, string duality, and nodal curves on K3’, Nuclear Phys. B 457 (1995), 484–512.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: 2
Total number of PDF views: 502 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 21st January 2021. This data will be updated every 24 hours.

Access
Open access
6 Cited by
Hostname: page-component-76cb886bbf-cdxmh Total loading time: 0.819 Render date: 2021-01-21T03:15:06.072Z Query parameters: { "hasAccess": "1", "openAccess": "1", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

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 KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES
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 KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES
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 KATZ–KLEMM–VAFA CONJECTURE FOR $K3$ SURFACES
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *