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Fano Shimura Varieties with Mostly Branched Cusps

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Birational Geometry, Kähler–Einstein Metrics and Degenerations (BGKEMD 2019, BGKEMD 2019, BGKEMD 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 409))

Abstract

We prove that the Satake-Baily-Borel compactification of certain Shimura varieties are Fano varieties, Calabi-Yau varieties or have ample canonical divisors with mild singularities. We also prove some variants statements, give applications and discuss various examples including new ones, for instance, the moduli spaces of unpolarized (log) Enriques surfaces.

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References

  1. Alexeev, V.: Log canonical singularities and complete moduli of stable pairs. arXiv:9608013

  2. Allcock, D., Freitag, E.: Cubic surfaces and Borcherds products. Comment. Math. Helv. 77(2), 270–296 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Allcock, D., Carlson, J.A., Toledo, D.: The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Algebr. Geom. 11, 659–724 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambro, F.: Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 220–239; translation in Proc. Steklov Inst. Math. 1(240), 214–233 (2003)

    Google Scholar 

  5. Ash, A., Mumford, D., Rapoport, M., Tai, Y.-S.: Smooth compactification of locally symmetric varieties. Cambridge Mathematical Library

    Google Scholar 

  6. Baily, W., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. 84(12), 442–528 (1966)

    Google Scholar 

  7. Barth, W., Nieto, I.: Abelian surfaces of type \((1,3)\) and quartic surfaces with \(16\) skew lines. J. Alg. Geom. 3, 173–222 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Behrens, N.: Singularities of ball quotients. Geom. Dedicata 159, 389–407 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Borcherds, R.: Automorphic forms on \(O_{s+2,2}(\mathbb{R} )\) and infinite products. Invent. Math. 120(1), 161–213 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Borcherds, R.: The moduli space of Enriques surfaces and the fake Monster Lie superalgebra. Topology 35(3), 699–710 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Borcherds, R.: Automorphic forms with singularities on Grassmannians. Invent. Math. 132(3), 491–562 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Braun, L., Greb, D., Langlois, K., Moraga, J.: Reductive quotients of KLT singularities. arXiv:2111.02812

  13. Bruinier, J.H.: Borcherds products on \(O(2, l)\) and Chern classes of Heegner divisors. In: Lecture Notes in Mathematics, vol. 1780. Springer, Berlin (2002)

    Google Scholar 

  14. Bruinier, J.H.: On the converse theorem for Borcherds products. J. Algbr. 397, 315–342 (2014)

    Google Scholar 

  15. Bruinier, J.H., Ehlen, S., Freitag, E.: Lattices with many Borcherds products. Math. Comp. 85(300), 1953–1981 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dai, X., Yoshikawa, K.: Analytic torsion for log-Enriques surfaces and Borcherds product. arXiv:2009.10302

  17. Dieckmann, C., Krieg, A., Woitalla, M.: The graded ring of modular forms on the Cayley half-space of degree two. Ramanujan J. 48(2), 385–398 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Drezet, J.-M., Narasimhan, M.S.: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math. 97(1), 53–94 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Freitag, E.: Siegelsche Modulfunktionen. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  20. Freitag, E., Salvati Manni, R.: Some Siegel threefolds with a Calabi-Yau model. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 833–850 (2010)

    Google Scholar 

  21. Fujino, O.: Some problems on Fano varieties. In: Proceeding to the Conference “Fukuso-kikagaku no shomondai (2010)” in Japanese

    Google Scholar 

  22. Fujino, O.: Fundamental theorems for the log minimal model program. Publ. Res. Inst. Math. Sci. 47(3), 727–789 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fujino, O.: Foundations of the minimal model program. Mathematical Society of Japan, MSJ Memoirs, Tokyo (2017)

    Book  MATH  Google Scholar 

  24. Fujino, O., Gongyo, Y.: On canonical bundle formulas and subadjunctions. Michigan Math. J. 61(2), 255–264 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gritsenko, V.: Reflective modular forms in algebraic geometry. arXiv:1012.4155

  26. Gritsenko, V.: \(24\) faces of the Borcherds modular forms \(\Phi _{12}\). arXiv:1203.6503

  27. Gritsenko, V.: Reflective modular forms and applications. Russ. Math. Surv. 73, 797–864 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gritsenko, V., Hulek, K.: Uniruledness of orthogonal modular varieties. J. Alg. Geom. 23, 711–725 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gritsenko, V., Hulek, K.: Moduli of polarized Enriques surfaces, in K3 surfaces and their moduli. Progr. Math. 315, 55–72 (2016)

    Google Scholar 

  30. Gritsenko, V., Hulek, K., Sankaran, G.: The Kodaira dimension of the moduli spaces of K3 surfaces. Invent. math. 169, 519–567 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gritsenko, V., Nikulin, V.: Automorphic forms and Lorentzian Kac-Moody algebras. I, Int. J. Math. 9(2), 153–199 (1998)

    Google Scholar 

  32. Gritsenko, V., Nikulin, V.: Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections. Proc. Lond. Math. Soc. 116(3), 485–533 (2018)

    Google Scholar 

  33. Hacon, C., Mckernan, J.: On Shokurov’s rational connectedness conjecture. Duke Math. J. 138(1), 119–136, 15 (2007)

    Google Scholar 

  34. Hashimoto, K., Ueda, K.: The ring of modular forms for the even unimodular lattice of signature \((2,10)\). Proc. Am. Math. Soc. 150(2), 547–558 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hofmann, E.: Borcherds products on unitary groups. Math. Ann. 358(3–4), 799–832 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Huybrechts, D.: Complete curves in moduli spaces of stable bundles on surfaces. Math. Ann. 298(1), 67–78 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ichikawa, T.: Siegel modular forms of degree \(2\) over rings. J. Number Theory 129(4), 818–823 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Igusa, J.: On Siegel modular forms of genus two. II, Am. J. Math. 86, 392–412 (1964)

    Google Scholar 

  39. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  40. Kondō, S.: On the Kodaira dimension of the moduli spaces of K3 surfaces. Compositio Math. 89, 251–299 (1993)

    MathSciNet  MATH  Google Scholar 

  41. Kondō, S.: The rationality of the moduli space of Enriques surfaces. Compositio Math. 91, 159–173 (1994)

    MathSciNet  MATH  Google Scholar 

  42. Kondō, S.: On the Kodaira dimension of the moduli spaces of K3 surfaces. II, Compositio Math. 116, 111–117 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kondō, S.: The moduli space of Enriques surfaces and Borcherds products. J. Alg. Geom. 11, 601–627 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  44. Lejarraga, P.: The moduli of Weierstrass fibrations over \(\mathbb{P} ^{1}\): rationality. Rocky Mountain J. Math. 23(2), 649–650 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  45. Li, J.: Kodaira dimension of moduli space of vector bundles on surfaces. Invent. Math. 115(1), 1–40 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  46. Looijenga, E.: The smoothing components of a triangle singularity. II, Math. Ann. 269(3), 357–387 (1984)

    Google Scholar 

  47. Ma, S.: The unirationality of the moduli spaces of 2-elementary K3 surfaces. With an appendix by Ken-Ichi Yoshikawa. Proc. Lond. Math. Soc. (3) 105(4), 757–786 (2012)

    Google Scholar 

  48. Ma, S.: On the Kodaira dimension of orthogonal modular varieties. Invent. Math. 859–911 (2018)

    Google Scholar 

  49. Maeda, Y.: Uniruledness of unitary Shimura varieties associated with Hermitian forms of signatures \((1,3), (1,4), (1,5)\). Master thesis (Department of Mathematics, Kyoto university). arXiv:2008.13106

  50. Maeda, Y.: Reflective obstructions of unitary modular varieties. arXiv:2204.01128

  51. Mumford, D.: Hirzebruch’s proportionality theorem in the noncompact case. Invent Math. 42, 239–277 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  52. Mumford, D.: On the Kodaira dimension of the Siegel modular variety. In: Algebraic Geometry-Open Problems. Lecture Notes in Mathematics, vol. 997, pp. 348–375. Springer, Berlin (1983)

    Google Scholar 

  53. Nagano, A., Ueda, K.: The ring of modular forms for the even unimodular lattice of signature \((2,18)\). Hiroshima Math. J. 52(1), 43–51 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  54. Namikawa, Y.: Periods of Enriques surfaces. Math. Ann. 270(2), 201–222 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  55. Odaka, Y., Spotti, C., Sun, S.: Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics. J. Differ. Geom. 102(1), 127–172 (2016)

    Article  MATH  Google Scholar 

  56. Odaka, Y., Oshima, Y.: Collapsing K3 Surface, Tropical Geometry and Moduli Compactifications of Satake, Morgan-Shalen Type, MSJ Memoirs. Mathematical Society of Japan, Tokyo (2021)

    MATH  Google Scholar 

  57. Odaka, Y.: PL density invariant for type II degenerating K3 surfaces, Moduli compactification and hyperKahler metrics. Nagoya Math, J. (2020)

    Google Scholar 

  58. Satake, I.: On compactifications of the quotient spaces for arithmetically defined discontinuous groups. Ann. Math. 72(2), 555–580 (1960)

    Google Scholar 

  59. Sterk, H.: Compactifications of the period space of Enriques surfaces I. Math. Z. 207(1), 1–36 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  60. Tai, Y.S.: On the Kodaira dimension of the moduli space of abelian varieties. Invent. Math. 68, 425–439 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  61. van der Geer, G.: On the geometry of a Siegel modular threefold. Math. Ann. 260(3), 317–350 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  62. van der Geer, G.: Siegel modular forms of degree two and three and invariant theory. arXiv:2102.02245

  63. Yoshikawa, K.: Calabi-Yau threefolds of Borcea-Voisin, analytic torsion, and Borcherds products. Astérisque No. 328(2009), 355–393 (2010)

    Google Scholar 

  64. Yoshikawa, K.: K3 surfaces with involution, equivalent analytic torsion, and automorphic forms on the moduli spaces, II: a structure theorem for \(r(M)>10\). J. Reine Angew. Math. 677, 15–70 (2013)

    Google Scholar 

  65. Wang, H., Williams, B.: Simple lattices and free algebras of modular forms. arXiv:2009.13343

  66. Wang, H., Williams, B.: Free algebras of modular forms on ball quotients. arXiv:2105.14892

  67. Zhang, D.-Q.: Logarithmic Enriques surfaces. J. Math. Kyoto Univ. 31, 419–466 (1991)

    Google Scholar 

  68. Zhang, Q.: Rational connectedness of log \(Q\)-Fano varieties. J. Reine Angew. Math. 590, 131–142 (2006)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

Y.M would like to express his gratitude to his adviser, Tetsushi Ito, for his helpful comments and warm encouragement. We would also like to thank Shouhei Ma for constructive discussion on quadratic and Hermitian lattices and thank Ken-ichi Yoshikawa for the discussion on modular forms. Y.M is supported by JST ACT-X JPMJAX200P. Y.O is partially supported by KAKENHI 18K13389, 20H00112 and 16H06335.

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Authors and Affiliations

  1. Advanced Research Laboratory, R&D Center, Sony Group Corporation, 1-7-1 Konan, Minato-ku, Tokyo, 108-0075, Japan

    Yota Maeda

  2. Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan

    Yota Maeda & Yuji Odaka

Authors
  1. Yota Maeda
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  2. Yuji Odaka
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Correspondence to Yuji Odaka .

Editor information

Editors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, UK

    Ivan Cheltsov

  2. Mathematics Department, Stony Brook University, Stony Brook, NY, USA

    Xiuxiong Chen

  3. Department of Mathematics, University of Miami, Coral Gables, FL, USA

    Ludmil Katzarkov

  4. Center for Geometry and Physics, Institute for Basic Science, Pohang, Korea (Republic of)

    Jihun Park

Appendix:   Matrix Definitions

Appendix:   Matrix Definitions

Appendix A: Matrix Definitions The following matrices are taken from [49, Appendix].

1.1 A.1   \(\mathbb {Q}(\sqrt{-1})\) Cases

A.1 \(\mathbb {Q}(\sqrt{-1})\) Cases Let \(\Lambda _{U\oplus U}\) be an even unimodular Hermitian lattice of signature (1, 1) over \(\mathcal {O}_{\mathbb {Q}(\sqrt{-1})}\) defined by the matrix

$$\begin{aligned} \frac{1}{2\sqrt{-1}} \begin{pmatrix} 0 &{} 1 \\ -1 &{} 0 \\ \end{pmatrix} \end{aligned}$$

whose associated quadratic lattice \((\Lambda _{U\oplus U})_Q\) is \(U\oplus U\).

Let \(\Lambda _{U\oplus U(2)}\) be an even Hermitian lattice of signature (1, 1) over \(\mathcal {O}_{\mathbb {Q}(\sqrt{-1})}\) defined by the matrix

$$\begin{aligned}\frac{1}{2} \begin{pmatrix} 0 &{} 1+\sqrt{-1} \\ 1-\sqrt{-1} &{} 0 \end{pmatrix} \end{aligned}$$

whose associated quadratic lattice \((\Lambda _{U\oplus U(2)})_Q\) is \(U\oplus U(2)\).

Let \(\Lambda _{E_8(-1)}\) be an even unimodular Hermitian lattice of signature (0, 4) over \(\mathcal {O}_{\mathbb {Q}(\sqrt{-1})}\) defined by the matrix

$$\begin{aligned}- \frac{1}{2}\begin{pmatrix} 2 &{} -\sqrt{-1} &{} -\sqrt{-1}&{}1\\ \sqrt{-1} &{} 2 &{} 1 &{} \sqrt{-1}\\ \sqrt{-1} &{} 1 &{} 2 &{} 1\\ 1&{} -\sqrt{-1} &{} 1 &{} 2\\ \end{pmatrix} \end{aligned}$$

whose associated quadratic lattice \((\Lambda _{E_8(-1)})_Q\) is \(E_8(-1)\). This matrix is called Iyanaga’s matrix.

1.2 A.2   \(\mathbb {Q}(\sqrt{-2})\) Cases

A.2 \(\mathbb {Q}(\sqrt{-2})\) Cases Let \(\Lambda '_{U\oplus U}\) be an even unimodular Hermitian lattice of signature (1, 1) over \(\mathcal {O}_{\mathbb {Q}(\sqrt{-2})}\) defined by the matrix

$$\begin{aligned} \frac{1}{2\sqrt{-2}} \begin{pmatrix} 0 &{} 1 \\ -1 &{} 0 \\ \end{pmatrix} \end{aligned}$$

whose associated quadratic lattice \((\Lambda '_{U\oplus U})_Q\) is \(U\oplus U\).

Let \(\Lambda '_{U\oplus U(2)}\) be a Hermitian lattice of signature (1, 1) over \(\mathcal {O}_{\mathbb {Q}(\sqrt{-2})}\) defined by the matrix

$$\begin{aligned} \begin{pmatrix} 0 &{} \frac{1}{2} \\ \frac{1 }{2} &{} 0 \end{pmatrix} \end{aligned}$$

whose associated quadratic lattice \((\Lambda '_{U\oplus U(2)})_Q\) is \(U\oplus U(2)\).

Let \(\Lambda '_{E_8(-1)}\) be an even unimodular Hermitian lattice of signature (0, 4) over \(\mathcal {O}_{\mathbb {Q}(\sqrt{-2})}\) defined by the matrix

$$\begin{aligned} -\frac{1}{2}\begin{pmatrix} 2 &{} 0 &{} \sqrt{-2}+1&{}\frac{1}{2}\sqrt{-2}\\ 0 &{} 2 &{} \frac{1}{2}\sqrt{-2} &{} 1-\sqrt{-2}\\ 1-\sqrt{-2} &{} -\frac{1}{2}\sqrt{-2} &{} 2 &{} 0\\ -\frac{1}{2}\sqrt{-2}&{} \sqrt{-2}+1 &{} 0 &{} 2\\ \end{pmatrix} \end{aligned}$$

whose associated quadratic lattice \((\Lambda '_{E_8(-1)})_Q\) is \(E_8(-1)\).

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Maeda, Y., Odaka, Y. (2023). Fano Shimura Varieties with Mostly Branched Cusps. In: Cheltsov, I., Chen, X., Katzarkov, L., Park, J. (eds) Birational Geometry, Kähler–Einstein Metrics and Degenerations. BGKEMD BGKEMD BGKEMD 2019 2019 2019. Springer Proceedings in Mathematics & Statistics, vol 409. Springer, Cham. https://doi.org/10.1007/978-3-031-17859-7_32

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