Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-04-28T15:36:05.383Z Has data issue: false hasContentIssue false
  • English
  • Français

Mirror symmetry and Hitchin system on Deligne–Mumford curves: Strominger–Yau–Zaslow duality

Part of: Surfaces and higher-dimensional varieties Curves Families, fibrations

Published online by Cambridge University Press:  06 May 2024

Yonghong Huang
Yonghong Huang*
Affiliation:
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, People’s Republic of China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We systematically study the moduli stacks of Higgs bundles, spectral curves, and Norm maps on Deligne–Mumford curves. As an application, under some mild conditions, we prove the Strominger–Yau–Zaslow duality for the moduli spaces of Higgs bundles over a hyperbolic stacky curve.

Keywords

Higgs bundlesDM curvesSYZ duality

MSC classification

Primary: 14D23: Stacks and moduli problems 14H40: Jacobians, Prym varieties 14J33: Mirror symmetry
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 58
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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.)

Article purchase

Temporarily unavailable

Footnotes

This work was partially supported by the Fundamental Research Funds for the Central Universities (Grant No. 34000-31610294) and the Xinjiang Key Laboratory of Applied Mathematics (Grant No. XJDX1401).

References

Abramovich, D., Graber, T., and Vistoli, A., Gromov–Witten theory of Deligne–Mumford stacks . Amer. J. Math. 130(2008), no. 5, 1337–1398.CrossRefGoogle Scholar
Adem, A. and Ruan, Y., Twisted orbifold K-theory . Commun. Math. Phys. 237(2003), no. 3, 533–556.CrossRefGoogle Scholar
Alper, J., Good moduli spaces for Artin stacks . Ann. Inst. Fourier (Grenoble) 63(2013), no. 6, 2349–2402.CrossRefGoogle Scholar
Aoki, M., Hom stacks . Manuscripta Math. 119(2006), no. 1, 37–56.CrossRefGoogle Scholar
Beauville, A., Narasimhan, M. S., and Ramanan, S., Spectral curves and the generalised theta divisor . J. Reine Angew. Math. 398(1989), 169–179.Google Scholar
Behrend, K., Introduction to algebraic stacks . In: Moduli spaces, London Mathematical Society Lecture Note Series, 411, Cambridge University Press, Cambridge, 2014, pp. 1–131.Google Scholar
Behrend, K. and Noohi, B., Uniformization of Deligne–Mumford curves . J. Reine Angew. Math. 599(2006), 111–153.Google Scholar
Biswas, I. and Dey, A., SYZ duality for parabolic Higgs moduli spaces . Nucl. Phys. B 862(2012), 327–340.CrossRefGoogle Scholar
Biswas, I., Majumder, S., and Wong, L., Parabolic Higgs bundles and $\varGamma$ -Higgs bundles . J. Aust. Math. Soc. 95(2013), no. 3, 315–328.CrossRefGoogle Scholar
Boden, H. and Yokogawa, K., Rationality of moduli spaces of parabolic bundles . J. Lond. Math. Soc. II. Ser. 59(1999), no. 2, 461–478.CrossRefGoogle Scholar
Borne, N., Fibrés paraboliques et champ des racines . Int. Math. Res. Not. 2007(2007), no. 16, Article no. rnm049, 38 pp.CrossRefGoogle Scholar
Brochard, S., Foncteur de Picard d’un champ algébrique . Math. Ann. 343(2009), no. 3, 541–602.CrossRefGoogle Scholar
Brochard, S., Finiteness theorems for the Picard objects of an algebraic stack . Adv. Math. 229(2012), no. 3, 1555–1585.CrossRefGoogle Scholar
Cadman, C., Using stacks to impose tangency conditions on curves . Amer. J. Math. 129(2007), no. 2, 405–427.CrossRefGoogle Scholar
Candelas, P., Ossa, X., Green, P., and Parkes, L., A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory . Nucl. Phys. B 359(1991), no. 1, 21–74.CrossRefGoogle Scholar
Donagi, R. and Pantev, T., Langlands duality for Hitchin systems . Invent. Math. 189(2012), 653–735.CrossRefGoogle Scholar
Fantechi, B., Gottsche, L., Illusie, L., Kleiman, S. L. and Nitsure, N., Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs, 123, American Mathematical Society, Providence, RI, 2005, x + 339 pp.Google Scholar
Fulton, W., Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2, Springer, Berlin, 1998, xiv + 470 pp.CrossRefGoogle Scholar
Gillet, H., Intersection theory on algebraic stacks and Q-varieties . J. Pure Appl. Algebra 34(1984), nos. 2–3, 193–240.CrossRefGoogle Scholar
Giraud, J., Cohomologie non Abélienne, Die Grundlehren der mathematischen Wissenschaften, 179, Springer, Berlin–New York, 1971, ix + 467 pp.CrossRefGoogle Scholar
Gothen, P. and Oliveira, A., Topological mirror symmetry for parabolic Higgs bundles . J. Geom. Phys. 137(2019), 7–34.CrossRefGoogle Scholar
Groechenig, M., Wyss, D., and Ziegler, P., Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration . Invent. Math. 221(2020), no. 2, 505–596.CrossRefGoogle Scholar
Grothendieck, A., Éléments de géométrie algébrique II: Étude globale élémentaire de quelques classes de morphismes . Publ. Math. Inst. Hautes Etudes Sci. 8(1961), 5–205.Google Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, New York–Heidelberg, 1977, xvi + 496 pp.CrossRefGoogle Scholar
Hausel, T. and Thaddeus, M., Mirror symmetry, Langlands duality, and the Hitchin system . Invent. Math. 153(2003), no. 1, 197–229.CrossRefGoogle Scholar
Hitchin, N. J., The self-duality equations on a Riemann surface . Proc. Lond. Math. Soc. 55(1987), no. 1, 59–126.CrossRefGoogle Scholar
Hitchin, N. J., Lectures on special Lagrangian submanifolds . In: Winter School on Mirror symmetry, vector bundles and Lagrangian submanifolds, (Cambridge, MA, 1999), American Mathematical Society, Providence, RI, 2001, pp. 151–182.Google Scholar
Hitchin, N. J., Langlands duality and ${G}_2$ spectral curves . Quart. J. Math. 58(2007), no. 3, 319–344.CrossRefGoogle Scholar
Huang, Y., Langton’s type theorem on algebraic orbifolds . Acta Math. Sin. 39(2023), 584–602. https://doi.org/10.1007/s10114-022-1185-4 CrossRefGoogle Scholar
Huybrechts, D. and Lehn, M., The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010, xviii + 325 pp.CrossRefGoogle Scholar
Huybrechts, D. and Schroeer, S., The Brauer group of analytic K3 surfaces . Int. Math. Res. Not. 50(2003), 2687–2698.CrossRefGoogle Scholar
Jiang, Y. and Kundu, P., The Tanaka–Thomas’s Vafa–Witten invariants via surface Deligne–Mumford stacks . Pure. Appl. Math. Q. 17(2021), no. 1, 503–573.CrossRefGoogle Scholar
Konno, H., Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface . J. Math. Soc. Japan 45(1993), 2253–2257.CrossRefGoogle Scholar
Kontsevich, M., Homological algebra of mirror symmetry . In: Proceedings of the international congress of mathematicians, ICM 94, August 3–11, 1994, Zürich, Switzerland. Vol. I, Birkhäuser, Basel, 1995, pp. 120–139.Google Scholar
Kresch, A., On the geometry of Deligne–Mumford stacks . In: Abramovich, D., Bertram, A., Katzarkov, L., Pandharipande, R., and Thaddeus, M. (eds.), Algebraic geometry: Seattle, American Mathematical Society, Providence, RI, 2005, pp. 259–271.Google Scholar
Kydonakis, G., Sun, H., and Zhao, L., Poisson structures on Moduli spaces of Higgs bundles over stacky curves. Adv. Geom. 24(2024), no. 2, 163–182.Google Scholar
Langton, S. G., Valuative criteria for families of vector bundles on algebraic varieties . Ann. of Math. 101(1975), no. 2, 88–110.CrossRefGoogle Scholar
Laszlo, Y., Linearizaton of group stack actions and the Picard group of the moduli of ${SL}_r/{\mu}_s$ -bundles on a curve . Bull. Soc. Math. France 125(1997), no. 4, 529–545.CrossRefGoogle Scholar
Lazarsfeld, R., Positivity in algebraic Geometry I. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 48, Springer, Berlin, 2004, xviii + 387 pp.Google Scholar
Lieblich, M., Twisted sheaves and the period-index problem . Compos. Math. 144(2008), no. 1, 1–31.CrossRefGoogle Scholar
Lieblich, M., Compactified moduli of projective bundles . Algebra Number Theory 3(2009), no. 6, 653–695.CrossRefGoogle Scholar
Logares, M. and Martens, J., Moduli of parabolic Higgs bundles and Atiyah algebroids . J. Reine Angew. Math. 649(2010), 89–116.Google Scholar
Marcolli, M. and Mathai, V., Twisted index theory on good orbifolds I: noncommutative Bloch theory . Comm. Cont. Math. I(1999), no. 4, 553–587.CrossRefGoogle Scholar
Markman, E., Spectral curves and integrable systems . Compos. Math. 93(1994), 255–290.Google Scholar
Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1986, 320 pp. Translated from the Japanese by M. Reid (English).Google Scholar
Maulik, D. and Shen, J., Endoscopic decompositions and the Hausel–Thaddeus conjecture . Forum Math. Pi 9(2021), e8–e49.CrossRefGoogle Scholar
McLean, R., Deformations of calibrated submanifolds . Commun. Anal. Geom. 6(1998), no. 4, 705–747.CrossRefGoogle Scholar
Milne, J., Étale cohomology, Princeton Mathematical Series, 33, Princeton University Press, Princeton, NJ, 1980, xiii + 323 pp.Google Scholar
Nasatyr, B. and Steer, B., Orbifold Riemann surfaces and the Yang–Mills–Higgs equations . Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22(1995), no. 4, 595–643.Google Scholar
Ngô, B., Fibration de Hitchin et endoscopie . Invent. Math. 164(2006), no. 2, 399–453.CrossRefGoogle Scholar
Ngô, B., Le lemme fondamental pour les algèbres de lie . Publ. Math. Inst. Hautes Études Sci. 111(2010), 1–169.CrossRefGoogle Scholar
Nironi, F., Moduli spaces of semistable sheaves on projective Deligne–Mumford stack. Preprint, arXiv:0811.1949v2.Google Scholar
Nitsure, N., Moduli space of semistable pairs on a curve . Proc. Lond. Math. Soc. 62(1991), no. 2, 275–300.CrossRefGoogle Scholar
Olsson, M., Algebraic spaces and stacks, American Mathematical Society Colloquium Publications, 62, American Mathematical Society, Providence, RI, 2016, xi + 298 pp.CrossRefGoogle Scholar
Olsson, M. and Starr, J., Quot functors for Deligne–Mumford stacks . Kleiman. Comm. Algebra 31 (2003), no. 8, 4069–4096. Special issue in honor of Steven L.CrossRefGoogle Scholar
Poma, F., Étale cohomology of a DM curve-stack with coefficients in ${G}_m$ . Monatsh. Math. 169(2013), no. 1, 33–50.CrossRefGoogle Scholar
Simpson, C., Local systems on proper algebraic V-manifolds . Pure Appl. Math. Q 7(2011), no. 4, 1675–1759. Special Issue: In memory of Eckart Viehweg.CrossRefGoogle Scholar
Strominger, A., Yau, S., and Zaslow, E., Mirror symmetry is T-duality . Nucl. Phys. 479(1996), 243–259.CrossRefGoogle Scholar
Tanaka, Y. and Thomas, R., Vafa–Witten invariants for projective surfaces I: stable case . J. Algebraic Geom. 29(2020), no. 4, 603–668.CrossRefGoogle Scholar
Vistoli, A., Intersection theory on algebraic stacks and on their moduli spaces . Invent. Math. 97(1989), no. 3, 613–670.CrossRefGoogle Scholar
Voight, J. and Zureick-Brown, D., The canonical ring of a Stacky curve, Memoirs of the American Mathematical Society, 277(1362), American Mathematical Society, Providence, RI, 2022, v + 144 pp.CrossRefGoogle Scholar
Yokogawa, K., Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves . J. Math. Kyoto Univ. 33(1993), no. 2, 451–504.Google Scholar