Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T13:47:26.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Threefolds fibred by mirror sextic double planes

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

Published online by Cambridge University Press:  24 June 2020

Remkes Kooistra  and
Alan Thompson
Remkes Kooistra
Affiliation:
The King’s University, 9125 – 50 St NW, Edmonton, AB T6B 2H3, Canada e-mail: [email protected]
Alan Thompson*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a systematic study of threefolds fibred by K3 surfaces that are mirror to sextic double planes. There are many parallels between this theory and the theory of elliptic surfaces. We show that the geometry of such threefolds is controlled by a pair of invariants, called the generalized functional and generalized homological invariants, and we derive an explicit birational model for them, which we call the Weierstrass form. We then describe how to resolve the singularities of the Weierstrass form to obtain the “minimal form”, which has mild singularities and is unique up to birational maps in codimension 2. Finally, we describe some of the geometric properties of threefolds in minimal form, including their singular fibres, canonical divisor, and Betti numbers.

Keywords

K3 surfacethreefoldfibration

MSC classification

Primary: 14D06: Fibrations, degenerations
Secondary: 14J30: $3$-folds 14J28: $K3$ surfaces and Enriques surfaces
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 5 , October 2021 , pp. 1305 - 1346
Check for updates
Copyright
© Canadian Mathematical Society 2020

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.)

References

Batyrev, V.V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3(1994), no. 3, 493–535.Google Scholar
Beukers, F. and Heckman, G., Monodromy for the hypergeometric function ${}_n{F}_{n-1}$ . Invent. Math. 95(1989), no. 2, 325–354. http://dx.doi.org/10.1007/BF01393900 CrossRefGoogle Scholar
Clingher, A. and Doran, C. F., Modular invariants for lattice polarized K3 surfaces . Michigan Math. J. 55(2007), no. 2, 355–393. http://dx.doi.org/10.1307/mmj/1187646999 CrossRefGoogle Scholar
Clingher, A., Doran, C. F., Lewis, J., and Whitcher, U., Normal forms, K3 surface moduli and modular parametrizations. In: Groups and symmetries, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009, pp. 81–98. http://dx.doi.org/10.1090/crmp/047/06 Google Scholar
Cox, D. A. and Katz, S., Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs, 68, American Mathematical Society, Providence, RI, 1999. http://dx.doi.org/10.1090/surv/068 CrossRefGoogle Scholar
del Angel, P. L., Müller-Stach, S., van Straten, D., and Zuo, K., Hodge classes associated to 1-parameter families of Calabi-Yau 3-folds. Acta Math. Vietnam 35(2010), no. 1, 7–22.Google Scholar
Decker, W., Greuel, G.-M., Pfister, G., and Schönemann, H., Singular 4-1-2—A computer algebra system for polynomial computations. 2019. http://www.singular.uni-kl.de Google Scholar
Doran, C. F., Harder, A., Novoseltsev, A. Y., and Thompson, A., Families of lattice polarized K3 surfaces with monodromy. Int. Math. Res. Notices (2015), no. 23, 12265–12318. http://dx.doi.org/10.1093/imrn/rnv071 Google Scholar
Doran, C. F., Harder, A., Novoseltsev, A. Y., and Thompson, A., Calabi-Yau threefolds fibred by Kummer surfaces associated to products of elliptic curves. In: String-Math 2014, Proc. Sympos. Pure Math., 93, Amer. Math. Soc., Providence, RI, 2016, pp. 263–287. http://dx.doi.org/10.1016/j.aim.2016.03.045 Google Scholar
Doran, C. F., Harder, A., Novoseltsev, A. Y., and Thompson, A., Calabi-Yau threefolds fibred by mirror quartic K3 surfaces. Adv. Math. 298(2016), 369–392. http://dx.doi.org/10.1016/j.aim.2016.03.045 CrossRefGoogle Scholar
Doran, C. F., Harder, A., Novoseltsev, A. Y., and Thompson, A., Calabi-Yau threefolds fibred by high rank lattice polarized K3 surfaces. Math. Z. 294(2020), nos. 1–2, 783–815. http://dx.doi.org/10.1007/s00209-019-02279-9 CrossRefGoogle Scholar
Doran, C. F. and Malmendier, A., Calabi-Yau manifolds realizing symplectically rigid monodromy tuples. Adv. Theor. Math. Phys. 23(2019), no. 5, 1271–1359. http://dx.doi.org/10.4310/atmp.2019.v23.n5.a3 CrossRefGoogle Scholar
Dolgachev, I. V., Mirror symmetry for lattice polarised K3 surfaces. J. Math. Sci. 81(1996), no. 3, 2599–2630.CrossRefGoogle Scholar
Friedman, R., Base change, automorphisms, and stable reduction for type III K3 surfaces. In: The birational geometry of degenerations (Cambridge, MA., 1981), Progr. Math., 29, Birkhäuser, Boston, MA, 1983, pp. 277–298.Google Scholar
Fujino, O., A canonical bundle formula for certain algebraic fiber spaces and its applications. Nagoya Math. J. 172(2003), 129–171. http://dx.doi.org/10.1017/S0027763000008679 CrossRefGoogle Scholar
Iliev, A., Katzarkov, L., and Przyjalkowski, V., Double solids, categories and non-rationality. Proc. Edinb. Math. Soc. (2) 57(2014), no. 1, 145–173. http://dx.doi.org/10.1017/S0013091513000898CrossRefGoogle Scholar
Kollár, J., Shafarevich maps and automorphic forms. M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. http://dx.doi.org/10.1515/978140086195 CrossRefGoogle Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, UK, 1998.CrossRefGoogle Scholar
Levelt, A. H. M., Hypergeometric functions. Ph.D. thesis, University of Amsterdam, Drukkerij Holland N.V., Amsterdam, 1961.Google Scholar
Nikulin, V. V., Finite automorphism groups of Kähler $K3$ surfaces. Trans. Moscow Math. Soc. 38(1980), no. 2, 71–135.Google Scholar
Nikulin, V. V., Integral symmetric bilinear forms and some of their applications . Math. USSR Izv. 14(1980), no. 1, 103–167.CrossRefGoogle Scholar
Nikulin, V. V., Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds. Mat. Sb. 208(2017), no. 7, 84–108. http://dx.doi.org/10.4213/sm8838Google Scholar
Nikulin, V. V., Toric Landau-Ginzburg models. Russ. Math. Surveys 73(2018), no. 6, 1033–1118.Google Scholar
Przyjalkowski, V., Weak Landau-Ginzburg models for smooth Fano threefolds. Izv. Math. 77(2013), no. 4, 772–794.CrossRefGoogle Scholar
Shioda, T. and Inose, H., On singular $K3$ surfaces. In: Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136.CrossRefGoogle Scholar
Smith, J., Picard-Fuchs differential equations for families of K3 surfaces. Ph.D. thesis, University of Warwick, July 2006.Google Scholar
Vaidya, A. M., An inequality for Euler’s totient function. Math. Stud. 35(1967), 79–80.Google Scholar
The Sage Developers, SageMath, the Sage Mathematics Software System (version 8.7), 2019. www.sagemath.orgGoogle Scholar
Voisin, C., Hodge theory and complex algebraic geometry. I. Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, UK, 2007.Google Scholar
Zucker, S., Hodge theory with degenerating coefficients: ${L}_2$ cohomology in the Poincaré metric. Ann. of Math. (2) 109(1979), no. 3, 415–476. http://dx.doi.org/10.2307/1971221CrossRefGoogle Scholar