Skip to main content
SpringerLink
Log in

Convergence of volume forms on a family of log Calabi–Yau varieties to a non-Archimedean measure

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We study the convergence of volume forms on a degenerating holomorphic family of log Calabi–Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let (X, B) be a holomorphic family of sub log canonical, log Calabi–Yau complex varieties parameterized by the punctured unit disk. Let \(\eta \) be a meromorphic form on X with poles along B such that the restriction of \(\eta \) is a top-dimensional form on each of the fibers. We show that the (possibly infinite) measures induced by the restriction of \(|\eta \wedge \overline{\eta }|\) to a fiber converge to a measure on the Berkovich analytification of \(X \setminus B\) as we approach the puncture. The convergence takes place on a hybrid space, which is obtained by filling in the space \(X \setminus B\) with the aforementioned Berkovich space over the puncture.

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

Similar content being viewed by others

References

  1. Baker, M., Payne, S., Rabinoff, J.: On the structure of non-Archimedean analytic curves. In: Tropical and Non-Archimedean Geometry, Contemp. Math., vol. 605, pp. 93–121. Amer. Math. Soc., Providence (2013). https://doi.org/10.1090/conm/605/12113

  2. Baker, M., Payne, S., Rabinoff, J.: Non-archimedean geometry, tropicalization, and metrics on curves. Algebr. Geom. 3(1), 63–105 (2016). https://doi.org/10.14231/AG-2016-004

  3. Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3(3), 493–535 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Berkovich, V.G.: Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33. American Mathematical Society, Providence (1990). https://doi.org/10.1090/surv/033

  5. Berkovich, V.G.: Étale cohomology for non-Archimedean analytic spaces. Publ. Math. l’IHÉS 78, 5–161 (1993). https://doi.org/10.1007/BF02712916

    Article  MATH  Google Scholar 

  6. Berkovich, V.G.: Smooth p-adic analytic spaces are locally contractible. Invent. Math. 137, 1–84 (1999). https://doi.org/10.1007/s002220050323

    Article  MathSciNet  MATH  Google Scholar 

  7. Berkovich, V.G.: Smooth \(p\)-adic analytic spaces are locally contractible. II. In: Geometric Aspects of Dwork Theory, vols. I, II, pp. 293–370. Walter de Gruyter, Berlin (2004)

  8. Berkovich, V.G.: A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures. In: Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin, vol. I. Progr. Math., vol. 269, pp. 49–67. Birkhäuser Boston, Inc., Boston (2009). https://doi.org/10.1007/978-0-8176-4745-2_2

  9. Boucksom, S., Favre, C., Jonsson, M.: Singular semipositive metrics in non-Archimedean geometry. J. Algebr. Geom. 25(1), 77–139 (2016). https://doi.org/10.1090/jag/656

    Article  MathSciNet  MATH  Google Scholar 

  10. Boucksom, S., Jonsson, M.: Tropical and non-Archimedean limits of degenerating families of volume forms. Journal de l’École polytechnique-Mathématiques 4, 87–139 (2017). https://doi.org/10.5802/jep.39

  11. Brown, M.V., Mazzon, E.: The essential skeleton of a product of degenerations. Compos. Math. 155(7), 1259–1300 (2019). https://doi.org/10.1112/s0010437x19007346

    Article  MathSciNet  MATH  Google Scholar 

  12. Chambert-Loir, A., Ducros, A.: Formes différentielles réelles et courants sur les espaces de Berkovich (2012). arXiv:1204.6277

  13. Clemens, C.H.: Degeneration of Kähler manifolds. Duke Math. J. 44(2), 215–290 (1977). http://projecteuclid.org/euclid.dmj/1077312231

  14. Favre, C.: Degeneration of endomorphisms of the complex projective space in the hybrid space. J. Inst. Math. Jussieu, 1–43 (2017). https://doi.org/10.1017/s147474801800035x

  15. Gubler, W., Rabinoff, J., Werner, A.: Skeletons and tropicalizations. Adv. Math. 294, 150–215 (2016). https://doi.org/10.1016/j.aim.2016.02.022

    Article  MathSciNet  MATH  Google Scholar 

  16. Jonsson, M., Nicaise, J.: Convergence of \(p\)-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces. J. Écol. Polytech. Math. 7, 287–336 (2020). https://doi.org/10.5802/jep.118

    Article  MathSciNet  MATH  Google Scholar 

  17. Kollár, J., Mori, S.: Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998). https://doi.org/10.1017/CBO9780511662560

  18. Kontsevich, M., Soibelman, Y.: Affine structures and non-Archimedean analytic spaces. In: The Unity of Mathematics, pp. 321–385. Springer, New York (2006). https://doi.org/10.1007/0-8176-4467-9_9

  19. Mandel, T.: Classification of rank 2 cluster varieties. SIGMA Symmetry Integrability. Geom. Methods Appl. 15, Paper 042, 32 (2019). https://doi.org/10.3842/SIGMA.2019.042

  20. Mustaţă, M., Nicaise, J.: Weight functions on non-Archimedean analytic spaces and the Kontsevich–Soibelman skeleton. Algebr. Geom. 2(3):365–404 (2015). https://doi.org/10.14231/AG-2015-016

  21. Odaka, Y.: Tropical geometric compactification of moduli, II: \(A_g\) case and holomorphic limits. International Mathematics Research Notices, 01 (2018). https://doi.org/10.1093/imrn/rnx293

  22. Pille-Schneider, L.: Hybrid convergence of Kähler–Einstein measures (2019). arXiv:1911.03357

  23. Temkin, M.: Metrization of differential pluriforms on Berkovich analytic spaces. In: Non-Archimedean and Tropical Geometry, Simons Symp., pp. 195–285. Springer, New York (2016)

  24. Tyomkin, I.: Tropical geometry and correspondence theorems via toric stacks. Math. Ann. 353(3), 945–995 (2012). https://doi.org/10.1007/s00208-011-0702-z

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I thank my advisor, Mattias Jonsson, for suggesting this problem, and also for his support and guidance. I also thank the anonymous referee for their comments and suggestions. This work was supported by the NSF grants DMS-1600011 and DMS-1900025.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Sanal Shivaprasad

Authors
  1. Sanal Shivaprasad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sanal Shivaprasad.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shivaprasad, S. Convergence of volume forms on a family of log Calabi–Yau varieties to a non-Archimedean measure. Math. Z. 301, 3849–3875 (2022). https://doi.org/10.1007/s00209-022-03032-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-022-03032-5

Search

Navigation