Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-09T08:14:00.680Z Has data issue: false hasContentIssue false
  • English
  • Français

DONALDSON–THOMAS INVARIANTS OF LOCAL ELLIPTIC SURFACES VIA THE TOPOLOGICAL VERTEX

Part of: Projective and enumerative geometry Cycles and subschemes

Published online by Cambridge University Press:  21 March 2019

JIM BRYAN  and
MARTIJN KOOL Open the ORCID record for MARTIJN KOOL [Opens in a new window]
JIM BRYAN
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2
MARTIJN KOOL
Affiliation:
Mathematical Institute, Utrecht University, Room 502, Budapestlaan 6, 3584 CD Utrecht, The Netherlands

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We compute the Donaldson–Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in Bryan et al. [‘Trace identities for the topological vertex’, Selecta Math. (N.S.)24 (2) (2018), 1527–1548, arXiv:math/1603.05271], we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz–Klemm–Vafa formula for primitive curve classes which is independent of the computation of Kawai–Yoshioka.

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 14C05: Parametrization (Chow and Hilbert schemes)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e7
Creative Commons
Creative Common License - CCCreative Common License - BY
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) 2019

References

Atiyah, M. F. and Macdonald, I. G., Introduction to commutative algebra (Addison-Wesley Publishing Co., Reading, MA-London-Do Mills, Ont., 1969).Google Scholar
Behrend, K., ‘Donaldson–Thomas type invariants via microlocal geometry’, Ann. of Math. (2) 170(3) (2009), 1307–1338. arXiv:math/0507523.Google Scholar
Behrend, K. and Fantechi, B., ‘Symmetric obstruction theories and Hilbert schemes of points on threefolds’, Algebra Number Theory 2(3) (2008), 313–345. arXiv:math/0512556.Google Scholar
Bridgeland, T., ‘Hall algebras and curve-counting invariants’, J. Amer. Math. Soc. 24(4) (2011), 969–998. arXiv:1002.4374.Google Scholar
Bryan, J., ‘The Donaldson–Thomas theory of K3 × E via the topological vertex’, inGeometry of Moduli, (eds. Christophersen, J. A. and Ranestad, K.) Abel Symposia, 14 (Springer, 2018).Google Scholar
Bryan, J., Kool, M. and Young, B., ‘Trace identities for the topological vertex’, Selecta Math. (N.S.) 24(2) (2018), 1527–1548. arXiv:math/1603.05271.Google Scholar
Bryan, J., Oberdieck, G., Pandharipande, R. and Yin, Q., ‘Curve counting on abelian surfaces and threefolds’, Algebraic Geometry (Compositio) 5(4) (2018), 398–463. arXiv:math/1506.00841.Google Scholar
Friedman, R. and Morgan, J., Smooth Four-manifolds and Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 27 (Springer, 1994).Google Scholar
Haiman, M., ‘ t, q-Catalan numbers and the Hilbert scheme’, Discrete Math. 193(1–3) (1998), 201–224. Selected papers in honor of Adriano Garsia (Taormina, 1994).Google Scholar
Kawai, T. and Yoshioka, K., ‘String partition functions and infinite products’, Adv. Theor. Math. Phys. 4(2) (2000), 397–485.Google Scholar
Kool, M. and Thomas, R., ‘Stable pairs with descendents on local surfaces I: the vertical component’, Pure Appl. Math. Q. 13(4) (2017), 581–638. arXiv:math/1605.02576.Google Scholar
MacPherson, R. D., ‘Chern classes for singular algebraic varieties’, Ann. of Math. (2) 100 (1974), 423–432.Google Scholar
Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R., ‘Gromov–Witten theory and Donaldson–Thomas theory. I’, Compos. Math. 142(5) (2006), 1263–1285. arXiv:math.AG/0312059.Google Scholar
Maulik, D., Pandharipande, R. and Thomas, R. P., ‘Curves on K3 surfaces and modular forms’, J. Topol. 3(4) (2010), 937–996. With an appendix by A. Pixton.Google Scholar
Miranda, R., The basic theory of elliptic surfaces. Dottorato di Ricerca in Matematica. [Doctorate in Mathematical Research]. ETS Editrice, Pisa, 1989. http://www.math.colostate.edu/∼miranda/BTES-Miranda.pdf.Google Scholar
Okounkov, A., Reshetikhin, N. and Vafa, C., ‘Quantum Calabi–Yau and classical crystals’, inThe Unity of Mathematics, Progress in Mathematics, 244 (Birkhäuser Boston, Boston, MA, 2006), 597–618. arXiv:hep-th/0309208.Google Scholar
Pandharipande, R. and Thomas, R. P., ‘The Katz–Klemm–Vafa conjecture for K3 surfaces’, Forum Math. Pi 4(e4) (2016), 111 arXiv:math/1404.6698.Google Scholar
Toda, Y., ‘Stability conditions and curve counting invariants on Calabi–Yau 3-folds’, Kyoto J. Math. 52(1) (2012), 1–50. arXiv:math/1103.4229.Google Scholar