Skip to main content
SpringerLink
Log in

Real loci in (log) Calabi–Yau manifolds via Kato–Nakayama spaces of toric degenerations

  • Research Article
  • Published:
European Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study the real loci of toric degenerations of complex varieties with reducible central fibre. We show that the topology of such degenerations can be explicitly described via the Kato–Nakayama space of the central fibre as a log space. We furthermore provide generalities of real structures in log geometry and their lift to Kato–Nakayama spaces. A key point of this paper is a description of the Kato–Nakayama space of a toric degeneration and its real locus, both as bundles determined by tropical data. We provide several examples including real toric degenerations of K3-surfaces and a toric degeneration of local \({\mathbb {P}}^2\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. The discussions in Sect. 6 on real structures in log geometry and in Sect. 8 on real toric degenerations also hold in the category of schemes over \({\mathbb {R}}\).

  2. See [17, Sections A.1 and A.2] for the translation to the earlier conventions of [18, 20].

  3. The discussion in [17] is on the complement of a part \({\widetilde{\Delta }}\subset B\) of the codimension two skeleton of the barycentric subdivision. There is a retraction of to a subset of \({\widetilde{\Delta }}\). However, the discussion on monomials works on any cell not contained in and with locally toroidal at some point of \(X_\tau \).

  4. This condition is non-trivial only if \(x\in \partial B\).

References

  1. Argüz, H.: Topological torus fibrations on Calabi–Yau manifolds via Kato–Nakayama spaces (2020). arXiv:2003.11589 (to appear in the Proceedings of the 26th Gökova Geometry Topology Conference)

  2. Argüz, H., Prince, T.: Real Lagrangians in Calabi–Yau threefolds. Proc. London Math. Soc. 121(2), 287–311 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Argüz, H., Prince, T.: On the cohomology groups of real Lagrangians in Calabi–Yau threefolds (2020). arXiv:2002.03957 (to appear in Experimental Mathematics)

  4. Bihan, F.: Asymptotic behaviour of Betti numbers of real algebraic surfaces. Comment. Math. Helv. 78(2), 227–244 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Castaño-Bernard, R., Matessi, D.: The fixed point set of anti-symplectic involutions of Lagrangian fibrations. Rend. Semin. Mat. Univ. Politec. Torino 68(3), 235–250 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Degtyarev, A.I., Kharlamov, V.M.: Topological properties of real algebraic varieties: du côté de chez Rokhlin. Russian Math. Surveys 55(4), 735–814 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Delaunay, C.: Real structures on smooth compact toric surfaces. In: Goldman, R., Krasuaskas, R. (eds.) Topics in Algebraic Geometry and Geometric Modeling. Contemporary Mathematics, vol. 334, pp. 267–290. American Mathematical Society, Providence (2003)

  8. Duistermaat, J.J.: On global action-angle coordinates. Comm. Pure Appl. Math. 33(6), 687–706 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Antisymplectic involution and Floer cohomology. Geom. Topol. 21(1), 1–106 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)

    Book  Google Scholar 

  11. Georgieva, P.: Open Gromov–Witten disk invariants in the presence of an anti-symplectic involution. Adv. Math. 301, 116–160 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gillam, W.D.: Oriented real blowup (2011). http://www.math.boun.edu.tr/instructors/wdgillam/orb.pdf

  13. Gross, M.: Topological mirror symmetry. Invent. Math. 144(1), 75–137 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gross, M.: Special Lagrangian fibrations I: topology. In: Saito, M.-H., et al. (eds.) Integrable Systems and Algebraic Geometry, pp. 156–193. World Scientific, River Edge (1998)

    Google Scholar 

  15. Gross, M.: Special Lagrangian fibrations. II. Geometry. In: Yau, S.-T. (ed.) Surveys in Differential Geometry: Differential Geometry Inspired by String Theory. Surveys in Differential Geometry, vol. 5, pp. 341–403. International Press, Boston (1999)

    Google Scholar 

  16. Gross, M., Hacking, P., Keel, S.: Mirror symmetry for log Calabi–Yau surfaces I. Publ. Math. Inst. Hautes Études Sci. 122, 65–168 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gross, M., Hacking, P., Siebert, B.: Theta functions on varieties with effective anti-canonical class (2016). arXiv:1601.07081v1

  18. Gross, M., Siebert, B.: Mirror symmetry via logarithmic degeneration data I. J. Differential Geom. 72(2), 169–338 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gross, M., Siebert, B.: Mirror symmetry via logarithmic degeneration data II. J. Algebraic Geom. 19(4), 679–780 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gross, M., Siebert, B.: From real affine to complex geometry. Ann. Math. 174(3), 1301–1428 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gross, M., Siebert, B.: An invitation to toric degenerations. In: Leung, N.C., Yau, S.-T. (eds.) Surveys in Differential Geometry, Vol. XVI. Geometry of Special Holonomy and Related Topics. Surveys in Differential Geometry, vol. 16, pp. 43–78. International Press, Somerville (2011)

    Google Scholar 

  22. Haase, C., Zharkov, I.: Integral affine structures on spheres: complete intersections. Int. Math. Res. Not. 2005(51), 3153–3167 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Haase, C., Zharkov, I.: Integral affine structures on spheres and torus fibrations of Calabi–Yau toric hypersurfaces I (2002). arXiv:0205321

  24. Haase, C., Zharkov, I.: Integral affine structures on spheres and torus fibrations of Calabi–Yau toric hypersurfaces II (2003). arXiv:0301222

  25. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

  26. Hatcher, A.E.: Concordance Spaces, Higher Simple-Homotopy Theory and Applications. In: Milgram, R.J. (ed.) Algebraic and Geometric Topology, Part I. Proceedings of Symposia in Pure Mathematics, vol. 32. American Mathematical Society, Providence (1978)

    Google Scholar 

  27. Huisman, J.: Real Abelian Varieties with Complex Mmultiplication. PhD thesis, Vrije Universiteit (1992)

  28. Itenberg, I.: Topology of real algebraic \(T\)-surfaces. Rev. Mat. Univ. Complut. Madrid 10(special issue), 131–152 (1997)

    MathSciNet  MATH  Google Scholar 

  29. Itenberg, I., Katzarkov, L., Mikhalkin, G., Zharkov, I.: Tropical homology. Math. Ann. 374(1–2), 963–1006 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Itenberg, I., Viro, O.: Asymptotically maximal real algebraic hypersurfaces of projective space. In: Proceedings of Gökova Geometry-Topology Conference, pp. 91–105. GGT, Gökova (2006)

  31. Kato, F.: Log smooth deformation theory. Tohoku Math. J. 48(3), 317–354 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kato, K.: Logarithmic structures of Fontaine-Illusie. In: Igusa, J.-I., et al. (eds.) Algebraic Analysis, Geometry, and Number Theory, pp. 191–224. Johns Hopkins University Press, Baltimore (1989)

    Google Scholar 

  33. Kato, K., Nakayama, C.: Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over \({ C}\). Kodai Math. J. 22(2), 161–186 (1999)

    MathSciNet  MATH  Google Scholar 

  34. Moishezon, B.: Complex Surfaces and Connected Sums of Complex Projective Planes. Lecture Notes in Mathematics, vol. 603. Springer, Berlin (1977)

  35. Moser-Jauslin, L., Terpereau, R.: Real structures on horospherical varieties. Michigan Math. J. (2020). https://doi.org/10.1307/mmj/20195793

  36. Nakayama, C., Ogus, A.: Relative rounding in toric and logarithmic geometry. Geom. Topol. 14(4), 2189–2241 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Nikulin, V.V.: Integral symmetric bilinear forms and some of their applications. Math. USSR-Izv. 14(1), 103–167 (1980)

    Article  MATH  Google Scholar 

  38. Nikulin, V.V., Saito, S.: Real \(K3\) surfaces with non-symplectic involution and applications. Proc. London Math. Soc. 90(3), 591–654 (2005)

    MathSciNet  MATH  Google Scholar 

  39. Ogus, A.: Lectures on Logarithmic Algebraic Geometry. Cambridge Studies in Advanced Mathematics, vol. 178. Cambridge University Press, Cambridge (2018)

    Book  MATH  Google Scholar 

  40. Pandharipande, R., Solomon, J., Walcher, J.: Disk enumeration on the quintic 3-fold. J. Amer. Math. Soc. 21(4), 1169–1209 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Parker, B.: Exploded manifolds. Adv. Math. 229(6), 3256–3319 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Renaudineau, A., Shaw, K.: Bounding the Betti numbers of real hypersurfaces near the tropical limit (2018). arXiv:1805.02030

  43. Ruan, W.-D.: Lagrangian torus fibration of quintic hypersurfaces. I. Fermat quintic case. In: Vafa, C., Yau, S.-T. (eds.) Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds. AMS/IP Studies in Advanced Mathematics, vol. 23, pp. 297–332. American Mathematical Society, Providence (2001)

    Google Scholar 

  44. Ruan, W.-D.: Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces. II. Technical results on gradient flow construction. J. Symplectic Geom. 1(3), 435–521 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ruan, W.-D.: Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces. III. Symplectic topological SYZ mirror construction for general quintics. J. Differential Geom. 63(2), 171–229 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ruddat, H., Sibilla, N., Treumann, D., Zaslow, E.: Skeleta of affine hypersurfaces. Geom. Topol. 18(3), 1343–1395 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ruddat, H., Siebert, B.: Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations (2019). arXiv:1907.03794

  48. Ruddat, H., Zharkov, I.: Compactifying torus fibrations over integral affine manifolds with singularities (2020). arXiv:2003.08521

  49. Ruddat, H., Zharkov, I.: Tailoring a pair of pants (2020). arXiv:2001.08267

  50. Siebenmann, L.C.: Deformation of homeomorphisms on stratified sets. I, II. Comment. Math. Helv. 47, 137–163 (1972)

    Article  MathSciNet  Google Scholar 

  51. Solomon, J.: Intersection Theory on the Moduli Space of Holomorphic Curves with Lagrangian Boundary Conditions. Ph.D. Thesis, Massachusetts Institute of Technology (2006). arXiv:math/0606429

  52. Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is \(T\)-duality. Nuclear Phys. B 479(1–2), 243–259 (1996)

    MathSciNet  MATH  Google Scholar 

  53. Viro, O.: Outline of results. http://www.math.stonybrook.edu/~oleg/math/research-st.html

Download references

Acknowledgements

This work was a part of my PhD thesis, which would not be possible without the support of my adviser Bernd Siebert. I am also indebted to Mark Gross for many useful conversations; a major part of this paper was written during a visit to the University of Cambridge hosted by him. I am grateful to Tom Coates and Dimitri Zvonkine for their useful suggestions which improved the exposition of the paper. Finally, many thanks to anonymous referees for their very useful feedback and corrections.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques, Université de Versailles St Quentin en Yvelines, 45 avenue des États-Unis, 78035, Versailles Cedex, France

    Hülya Argüz

Authors
  1. Hülya Argüz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hülya Argüz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This project has received funding from the “Research Training Group 1670 Mathematics inspired by String Theory” at Universität Hamburg funded by the Deutsche Forschungsgemeinschaft (DFG), the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 682603), and from Fondation Mathématiques Jacques Hadamard.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Argüz, H. Real loci in (log) Calabi–Yau manifolds via Kato–Nakayama spaces of toric degenerations. European Journal of Mathematics 7, 869–930 (2021). https://doi.org/10.1007/s40879-021-00454-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40879-021-00454-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation