On deformations of ℚ-Fano 3-folds

Sano, Taro

doi:10.1090/jag/672

On deformations of $\mathbb {Q}$-Fano $3$-folds

Author: Taro Sano
Journal: J. Algebraic Geom. 25 (2016), 141-176
DOI: https://doi.org/10.1090/jag/672
Published electronically: August 4, 2015
MathSciNet review: 3419958
Full-text PDF

Abstract | References | Additional Information

Abstract: We study the deformation theory of a $\mathbb {Q}$-Fano 3-fold with only terminal singularities. First, we show that the Kuranishi space of a $\mathbb {Q}$-Fano 3-fold is smooth. Second, we show that every $\mathbb {Q}$-Fano 3-fold with only “ordinary” terminal singularities is $\mathbb {Q}$-smoothable; that is, it can be deformed to a $\mathbb {Q}$-Fano 3-fold with only quotient singularities. Finally, we prove $\mathbb {Q}$-smoothability of a $\mathbb {Q}$-Fano 3-fold assuming the existence of a Du Val anticanonical element. As an application, we get the genus bound for primary $\mathbb {Q}$-Fano 3-folds with Du Val anticanonical elements.

References

Dan Abramovich and Jianhua Wang, Equivariant resolution of singularities in characteristic $0$, Math. Res. Lett. 4 (1997), no. 2-3, 427–433. MR 1453072, DOI https://doi.org/10.4310/MRL.1997.v4.n3.a11
Selma Altınok, Gavin Brown, and Miles Reid, Fano 3-folds, $K3$ surfaces and graded rings, Topology and geometry: commemorating SISTAG, Contemp. Math., vol. 314, Amer. Math. Soc., Providence, RI, 2002, pp. 25–53. MR 1941620, DOI https://doi.org/10.1090/conm/314/05420
D. M. Burns Jr. and Jonathan M. Wahl, Local contributions to global deformations of surfaces, Invent. Math. 26 (1974), 67–88. MR 349675, DOI https://doi.org/10.1007/BF01406846
Barbara Fantechi and Marco Manetti, Obstruction calculus for functors of Artin rings. I, J. Algebra 202 (1998), no. 2, 541–576. MR 1617687, DOI https://doi.org/10.1006/jabr.1997.7239
William Fulton, On the topology of algebraic varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 15–46. MR 927947
Daniel Greb and Sönke Rollenske, Torsion and cotorsion in the sheaf of Kähler differentials on some mild singularities, Math. Res. Lett. 18 (2011), no. 6, 1259–1269. MR 2915479, DOI https://doi.org/10.4310/MRL.2011.v18.n6.a14
Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620
Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
Robin Hartshorne, Deformation theory, Graduate Texts in Mathematics, vol. 257, Springer, New York, 2010. MR 2583634
János Kollár, Flatness criteria, J. Algebra 175 (1995), no. 2, 715–727. MR 1339664, DOI https://doi.org/10.1006/jabr.1995.1209
János Kollár and Shigefumi Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), no. 3, 533–703. MR 1149195, DOI https://doi.org/10.1090/S0894-0347-1992-1149195-9
János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959
Ernst Kunz, Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 864975
Tatsuhiro Minagawa, Deformations of ${\bf Q}$-Calabi-Yau $3$-folds and ${\bf Q}$-Fano $3$-folds of Fano index $1$, J. Math. Sci. Univ. Tokyo 6 (1999), no. 2, 397–414. MR 1707207
Tatsuhiro Minagawa, Deformations of weak Fano 3-folds with only terminal singularities, Osaka J. Math. 38 (2001), no. 3, 533–540. MR 1860839
Shigefumi Mori, On $3$-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43–66. MR 792770, DOI https://doi.org/10.1017/S0027763000021358
Shigeru Mukai, New developments in the theory of Fano threefolds: vector bundle method and moduli problems [translation of Sūgaku 47 (1995), no. 2, 125–144; MR1364825 (96m:14059)], Sugaku Expositions 15 (2002), no. 2, 125–150. Sugaku expositions. MR 1944132
Yoshinori Namikawa, On deformations of Calabi-Yau $3$-folds with terminal singularities, Topology 33 (1994), no. 3, 429–446. MR 1286924, DOI https://doi.org/10.1016/0040-9383%2894%2990021-3
Yoshinori Namikawa, Smoothing Fano $3$-folds, J. Algebraic Geom. 6 (1997), no. 2, 307–324. MR 1489117
Yoshinori Namikawa, Calabi-Yau threefolds and deformation theory [translation of Sūgaku 48 (1996), no. 4, 337–357; MR1614448 (2000h:14032)], Sugaku Expositions 15 (2002), no. 1, 1–29. Saguku Expositions. MR 1898900
Yoshinori Namikawa and J. H. M. Steenbrink, Global smoothing of Calabi-Yau threefolds, Invent. Math. 122 (1995), no. 2, 403–419. MR 1358982, DOI https://doi.org/10.1007/BF01231450
Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures, 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], vol. 52, Springer-Verlag, Berlin, 2008. MR 2393625
G. V. Ravindra and V. Srinivas, The Grothendieck-Lefschetz theorem for normal projective varieties, J. Algebraic Geom. 15 (2006), no. 3, 563–590. MR 2219849, DOI https://doi.org/10.1090/S1056-3911-05-00421-2
G. V. Ravindra and V. Srinivas, The Noether-Lefschetz theorem for the divisor class group, J. Algebra 322 (2009), no. 9, 3373–3391. MR 2567426, DOI https://doi.org/10.1016/j.jalgebra.2008.09.003

M. Reid, Projective morphisms according to Kawamata, Warwick preprint, 1983, www.maths.warwick.ac.uk/~miles/3folds/Ka.pdf

Miles Reid, Minimal models of canonical $3$-folds, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 131–180. MR 715649, DOI https://doi.org/10.2969/aspm/00110131

Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414. MR 927963

Michael Schlessinger, Rigidity of quotient singularities, Invent. Math. 14 (1971), 17–26. MR 292830, DOI https://doi.org/10.1007/BF01418741

Edoardo Sernesi, Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 334, Springer-Verlag, Berlin, 2006. MR 2247603

E. Sernesi, Errata and Addenda to “Deformations of algebraic schemes”, www.mat. uniroma3.it/users/sernesi/errataDAS.pdf

V. V. Šokurov, Smoothness of a general anticanonical divisor on a Fano variety, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 430–441 (Russian). MR 534602

Hiromichi Takagi, Classification of primary $\Bbb Q$-Fano threefolds with anti-canonical Du Val $K3$ surfaces. I, J. Algebraic Geom. 15 (2006), no. 1, 31–85. MR 2177195, DOI https://doi.org/10.1090/S1056-3911-05-00416-9

Hiromichi Takagi, On classification of $\Bbb Q$-Fano 3-folds of Gorenstein index 2. I, II, Nagoya Math. J. 167 (2002), 117–155, 157–216. MR 1924722, DOI https://doi.org/10.1017/S0027763000025460

Jonathan M. Wahl, Equisingular deformations of normal surface singularities. I, Ann. of Math. (2) 104 (1976), no. 2, 325–356. MR 422270, DOI https://doi.org/10.2307/1971049

Additional Information

Taro Sano
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom – and – Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
Email: tarosano222@gmail.com

Received by editor(s): November 25, 2012
Received by editor(s) in revised form: July 15, 2013, March 14, 2014, March 21, 2014, August 5, 2014, and September 2, 2014
Published electronically: August 4, 2015
Additional Notes: The author was partially supported by a Warwick Postgraduate Research Scholarship. He was partially funded by the Korean government WCU Grant R33-2008-000-10101-0, Research Institute for Mathematical Sciences and Higher School of Economics
Dedicated: Dedicated to Professor Yujiro Kawamata on the occasion of his 60th birthday
Article copyright: © Copyright 2015 University Press, Inc.