Skip to main content Accessibility help
×
Home

Moduli spaces of stable quotients and wall-crossing phenomena

  • Yukinobu Toda (a1)

Abstract

The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich’s stable map compactification and Marian–Oprea–Pandharipande’s stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck’s Quot scheme. In this paper, we give the notion of ‘ ϵ-stable quotients’ for a positive real number ϵ, and show that stable maps and stable quotients are related by wall-crossing phenomena. We will also discuss Gromov–Witten type invariants associated to ϵ-stable quotients, and investigate them under wall crossing.

    • 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.

      Moduli spaces of stable quotients and wall-crossing phenomena
      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.

      Moduli spaces of stable quotients and wall-crossing phenomena
      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.

      Moduli spaces of stable quotients and wall-crossing phenomena
      Available formats
      ×

Copyright

References

Hide All
[AMV04]Aganagic, M., Marino, M. and Vafa, C., All loop topological string amplitudes from Chern–Simons theory, Comm. Math. Phys. 247 (2004), 467512.
[AG08]Alexeev, V. and Guy, M., Moduli of weighted stable maps and their gravitational descendants, J. Inst. Math. Jussieu 7 (2008), 425456.
[BM09]Bayer, A. and Macri, E., The space of stability conditions on the local projective plane, Preprint, arXiv:0912.0043.
[Beh97]Behrend, K., Gromov–Witten invariants in algebraic geometry, Invent. Math. 127 (1997), 601617.
[BF97]Behrend, K. and Fantechi, B., The intrinsic normal cone, Invent. Math. 128 (1997), 4588.
[Ber94]Bertram, A., Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian, Internat. J. Math. 5 (1994), 811825.
[Ber97]Bertram, A., Quantum Schubert calculus, Adv. Math. 128 (1997), 289305.
[BDW96]Bertram, A., Daskalopoulos, G. and Wentworth, R., Gromov invariants for holomorphic maps from Riemann surfaces to Grasmannians, J. Amer. Math. Soc. 9 (1996), 529571.
[Bri10]Bridgeland, T., Hall algebras and curve-counting invariants, Preprint, arXiv:1002.4374.
[CK09]Ciocan-Fontanine, I. and Kapranov, M., Virtual fundamental classes via dg-manifolds, Geom. Topol. 13 (2009), 17791804.
[FP00]Faber, C. and Pandharipande, R., Hodge integrals and Gromov–Witten theory, Invent. Math. 139 (2000), 173199.
[GP99]Graber, T. and Pandharipande, R., Localization of virtual classes, Invent. Math. 135 (1999), 487518.
[Has03]Hassett, B., Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), 316352.
[JS08]Joyce, D. and Song, Y., A theory of generalized Donaldson–Thomas invariants, Preprint, arXiv:0810.5645.
[KMM87]Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, Adv. Stud. Pure Math. 10 (1987), 283360.
[KM10]Kiem, Y. H. and Moon, H. B., Moduli spaces of weighted pointed stable rational curves via GIT, Preprint, arXiv:1002.2461.
[KP01]Kim, B. and Pandharipande, R., The connectedness of the moduli spaces of maps to homogeneous spaces, in Symplectic geometry and mirror symmetry (World Scientific Publishing, River Edge, NJ, 2001), 187201.
[Kir85]Kirwan, F., Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math. (2) 122 (1985), 4185.
[KM98]Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).
[Kon95]Kontsevich, M., Enumeration of rational curves via torus actions. The moduli space of curves, Progr. Math. 129 (1995), 335368.
[KM94]Kontsevich, M. and Manin, Y., Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525562.
[KS08]Kontsevich, M. and Soibelman, Y., Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, Preprint, arXiv:0811.2435.
[LM00]Laumon, G. and Moret-Bailly, L., Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 39 (Springer, Berlin, 2000).
[LT98]Li, J. and Tian, G., Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119174.
[MO07]Marian, A. and Oprea, D., Virtual intersections on the Quot schemes and Vafa–Intriligator formulas, Duke Math. J. 136 (2007), 81113.
[MOP09]Marian, A., Oprea, D. and Pandharipande, R., The moduli space of stable quotients, Preprint, arXiv:0904.2992.
[MNOP06]Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R., Gromov–Witten theory and Donaldson–Thomas theory. I, Compositio Math. 142 (2006), 12631285.
[MFK94]Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, third enlarged edition (Springer, Berlin, 1994).
[MM07]Mustatǎ, A. and Mustatǎ, A., Intermediate moduli spaces of stable maps, Invent. Math. 167 (2007), 4790.
[PR03]Popa, M. and Roth, M., Stable maps and Quot schemes, Invent. Math. 152 (2003), 625663.
[ST09]Stoppa, J. and Thomas, R. P., Hilbert schemes and stable pairs: GIT and derived category wall crossings, Preprint, arXiv:0903.1444.
[Tho00]Thomas, R. P., A holomorphic Casson invariant for Calabi–Yau 3-folds and bundles on K3-fibrations, J. Differential Geom. 54 (2000), 367438.
[Tod10a]Toda, Y., Curve counting theories via stable objects I: DT/PT correspondence, J. Amer. Math. Soc. 23 (2010), 11191157.
[Tod10b]Toda, Y., Generating functions of stable pair invariants via wall-crossings in derived categories, in New developments in algebraic geometry, integrable systems and mirror symmetry, RIMS, Kyoto, 2008, Advanced Studies in Pure Mathematics, vol. 59 (Mathematical Society of Japan, Tokyo, 2010), 389434.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Related content

Powered by UNSILO

Moduli spaces of stable quotients and wall-crossing phenomena

  • Yukinobu Toda (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.