Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension
Authors:
Sándor J Kovács and Zsolt Patakfalvi
Journal:
J. Amer. Math. Soc. 30 (2017), 959-1021
MSC (2010):
Primary 14J10
DOI:
https://doi.org/10.1090/jams/871
Published electronically:
December 15, 2016
MathSciNet review:
3671934
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We prove that any coarse moduli space of stable log-varieties of general type is projective. We also prove subadditivity of log-Kodaira dimension for fiber spaces whose general fiber is of log general type.
- Dan Abramovich, A high fibered power of a family of varieties of general type dominates a variety of general type, Invent. Math. 128 (1997), no. 3, 481–494. MR 1452430, DOI https://doi.org/10.1007/s002220050149
- Dan Abramovich and Brendan Hassett, Stable varieties with a twist, Classification of algebraic varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 1–38. MR 2779465, DOI https://doi.org/10.4171/007-1/1
- Valery Alexeev, Boundedness and $K^2$ for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779–810. MR 1298994, DOI https://doi.org/10.1142/S0129167X94000395
- Valery Alexeev, Moduli spaces $M_{g,n}(W)$ for surfaces, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 1–22. MR 1463171, DOI https://doi.org/10.1006/jcat.1996.0357
- Kenneth Ascher and Amos Turchet, A fibered power theorem for pairs of log general type, Algebra Number Theory 10 (2016), no. 7, 1581–1600. MR 3554241, DOI https://doi.org/10.2140/ant.2016.10.1581
- W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574
- Caucher Birkar, The Iitaka conjecture $C_{n,m}$ in dimension six, Compos. Math. 145 (2009), no. 6, 1442–1446. MR 2575089, DOI https://doi.org/10.1112/S0010437X09004187
- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039, DOI https://doi.org/10.1090/S0894-0347-09-00649-3
- Paolo Bravi, Jacopo Gandini, Andrea Maffei, and Alessandro Ruzzi, Normality and non-normality of group compactifications in simple projective spaces, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 6, 2435–2461 (2012) (English, with English and French summaries). MR 2976317, DOI https://doi.org/10.5802/aif.2679
- J. Cao and M. Păun: Kodaira dimension of algebraic fiber spaces over abelian varieties, Invent. Math. (2016), doi:10.1007/s00222-016-0672-6.
- Jungkai Alfred Chen and Christopher D. Hacon, Kodaira dimension of irregular varieties, Invent. Math. 186 (2011), no. 3, 481–500. MR 2854084, DOI https://doi.org/10.1007/s00222-011-0323-x
- Y. Chen and L. Zhang: The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics, arXiv:1305.6024 (2013).
- Brian Conrad, Grothendieck duality and base change, Lecture Notes in Mathematics, vol. 1750, Springer-Verlag, Berlin, 2000. MR 1804902
- B. Conrad: The Keel-Mori theorem via stacks, http://math.stanford.edu/~conrad/papers/coarsespace.pdf (2005).
- Corrado De Concini, Normality and non normality of certain semigroups and orbit closures, Algebraic transformation groups and algebraic varieties, Encyclopaedia Math. Sci., vol. 132, Springer, Berlin, 2004, pp. 15–35. MR 2090668, DOI https://doi.org/10.1007/978-3-662-05652-3_3
- P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240
- Hélène Esnault and Eckart Viehweg, Effective bounds for semipositive sheaves and for the height of points on curves over complex function fields, Compositio Math. 76 (1990), no. 1-2, 69–85. Algebraic geometry (Berlin, 1988). MR 1078858
- Hubert Flenner, Die Sätze von Bertini für lokale Ringe, Math. Ann. 229 (1977), no. 2, 97–111 (German). MR 460317, DOI https://doi.org/10.1007/BF01351596
- Osamu Fujino, Algebraic fiber spaces whose general fibers are of maximal Albanese dimension, Nagoya Math. J. 172 (2003), 111–127. MR 2019522, DOI https://doi.org/10.1017/S0027763000008667
- O. Fujino: Semi-positivity theorems for moduli problems, arXiv:1210.5784 (2012).
- Osamu Fujino, On maximal Albanese dimensional varieties, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 8, 92–95. MR 3127923, DOI https://doi.org/10.3792/pjaa.89.92
- O. Fujino: Notes on the weak positivity theorems, arXiv:1406.1834 (2014).
- O. Fujino: On subadditivity of the logarithmic Kodaira dimension, arXiv:1406.2759 (2014).
- O. Fujino: Subadditivity of the logarithmic Kodaira dimension for morphisms of relative dimension one revisited, https://www.math.kyoto-u.ac.jp/~fujino/revisited2015-2.pdf (2015).
- D. Gieseker, Global moduli for surfaces of general type, Invent. Math. 43 (1977), no. 3, 233–282. MR 498596, DOI https://doi.org/10.1007/BF01390081
- A. Grothendieck: Éléments de géométrie algébrique IV., Inst. Hautes Études Sci. Publ. Math. (1964–1967), no. 20, 24, 28, 32.
- C. D. Hacon, J. McKernan, and C. Xu: Boundedness of moduli of varieties of general type, arXiv:1412.1186 (2014).
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Robin Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. MR 597077, DOI https://doi.org/10.1007/BF01467074
- Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352. MR 1957831, DOI https://doi.org/10.1016/S0001-8708%2802%2900058-0
- Brendan Hassett and Donghoon Hyeon, Log canonical models for the moduli space of curves: the first divisorial contraction, Trans. Amer. Math. Soc. 361 (2009), no. 8, 4471–4489. MR 2500894, DOI https://doi.org/10.1090/S0002-9947-09-04819-3
- Brendan Hassett and Donghoon Hyeon, Log minimal model program for the moduli space of stable curves: the first flip, Ann. of Math. (2) 177 (2013), no. 3, 911–968. MR 3034291, DOI https://doi.org/10.4007/annals.2013.177.3.3
- Brendan Hassett and Sándor J. Kovács, Reflexive pull-backs and base extension, J. Algebraic Geom. 13 (2004), no. 2, 233–247. MR 2047697, DOI https://doi.org/10.1090/S1056-3911-03-00331-X
- Shigeru Iitaka, Algebraic geometry, Graduate Texts in Mathematics, vol. 76, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties; North-Holland Mathematical Library, 24. MR 637060
- Masayuki Kawakita, Inversion of adjunction on log canonicity, Invent. Math. 167 (2007), no. 1, 129–133. MR 2264806, DOI https://doi.org/10.1007/s00222-006-0008-z
- Yujiro Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253–276. MR 622451
- Yujiro Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46. MR 814013, DOI https://doi.org/10.1515/crll.1985.363.1
- Seán Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149 (1999), no. 1, 253–286. MR 1680559, DOI https://doi.org/10.2307/121025
- Seán Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193–213. MR 1432041, DOI https://doi.org/10.2307/2951828
- Finn F. Knudsen, The projectivity of the moduli space of stable curves. III. The line bundles on $M_{g,n}$, and a proof of the projectivity of $\overline M_{g,n}$ in characteristic $0$, Math. Scand. 52 (1983), no. 2, 200–212. MR 702954, DOI https://doi.org/10.7146/math.scand.a-12002
- J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299–338. MR 922803, DOI https://doi.org/10.1007/BF01389370
- János Kollár, Subadditivity of the Kodaira dimension: fibers of general type, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 361–398. MR 946244, DOI https://doi.org/10.2969/aspm/01010361
- János Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. MR 1064874
- János Kollár, Rational curves on algebraic varieties, 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. 32, Springer-Verlag, Berlin, 1996. MR 1440180
- J. Kollár: Hulls and husks, arXiv:0805.0576 (2008).
- János Kollár, Moduli of varieties of general type, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 131–157. MR 3184176
- János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950
- J. Kollár: Moduli of higher dimensional varieties: book in preparation, manuscript (2014).
- János Kollár and Sándor J. Kovács, Log canonical singularities are Du Bois, J. Amer. Math. Soc. 23 (2010), no. 3, 791–813. MR 2629988, DOI https://doi.org/10.1090/S0894-0347-10-00663-6
- 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
- Ching-Jui Lai, Varieties fibered by good minimal models, Math. Ann. 350 (2011), no. 3, 533–547. MR 2805635, DOI https://doi.org/10.1007/s00208-010-0574-7
- G. Laumon and L. Moret-Bailly: Champs algébriques, 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. 39, Springer-Verlag, Berlin, 2000.
- Robert Lazarsfeld, Positivity in algebraic geometry. I, 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. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
- Robert Lazarsfeld, Positivity in algebraic geometry. II, 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. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472
- Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208
- Zsolt Patakfalvi, Semi-positivity in positive characteristics, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 5, 991–1025 (English, with English and French summaries). MR 3294622, DOI https://doi.org/10.24033/asens.2232
- Zsolt Patakfalvi, Fibered stable varieties, Trans. Amer. Math. Soc. 368 (2016), no. 3, 1837–1869. MR 3449226, DOI https://doi.org/10.1090/S0002-9947-2015-06386-7
- Zs. Patakfalvi: On subadditivity of Kodaira dimension in positive characteristic over a general type base, to appear in the Journal of Algebraic Geometry (2016).
- Zsolt Patakfalvi and Karl Schwede, Depth of $F$-singularities and base change of relative canonical sheaves, J. Inst. Math. Jussieu 13 (2014), no. 1, 43–63. MR 3134015, DOI https://doi.org/10.1017/S1474748013000066
- Zs. Patakfalvi and C. Xu: Ampleness of the CM line bundle on the moduli space of canonically polarized varieties, to appear in Algebraic Geometry (2015).
- N. I. Shepherd-Barron, Degenerations with numerically effective canonical divisor, The birational geometry of degenerations (Cambridge, Mass., 1981) Progr. Math., vol. 29, Birkhäuser Boston, Boston, MA, 1983, pp. 33–84. MR 690263
- Stacks Project Authors: Stacks project, http://stacks.math.columbia.edu.
- D. A. Timashëv, Equivariant compactifications of reductive groups, Mat. Sb. 194 (2003), no. 4, 119–146 (Russian, with Russian summary); English transl., Sb. Math. 194 (2003), no. 3-4, 589–616. MR 1992080, DOI https://doi.org/10.1070/SM2003v194n04ABEH000731
- Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR 0506253
- Eckart Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 329–353. MR 715656, DOI https://doi.org/10.2969/aspm/00110329
- Eckart Viehweg, Weak positivity and the additivity of the Kodaira dimension. II. The local Torelli map, Classification of algebraic and analytic manifolds (Katata, 1982) Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 567–589. MR 728619
- Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632
- Eckart Viehweg and Kang Zuo, Base spaces of non-isotrivial families of smooth minimal models, Complex geometry (Göttingen, 2000) Springer, Berlin, 2002, pp. 279–328. MR 1922109
- Xiaowei Wang and Chenyang Xu, Nonexistence of asymptotic GIT compactification, Duke Math. J. 163 (2014), no. 12, 2217–2241. MR 3263033, DOI https://doi.org/10.1215/00127094-2785806
Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 14J10
Retrieve articles in all journals with MSC (2010): 14J10
Additional Information
Sándor J Kovács
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
MR Author ID:
289685
Email:
skovacs@uw.edu
Zsolt Patakfalvi
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544-1000, USA
Address at time of publication:
EPFL, SB MATHGEOM CAG MA, B3 444 (Bâtiment MA), Station 8, CH-1015, Lausanne, Switzerland
Email:
zsolt.patakfalvi@epfl.ch
Received by editor(s):
July 10, 2015
Received by editor(s) in revised form:
February 10, 2016, and July 20, 2016
Published electronically:
December 15, 2016
Additional Notes:
The first author was supported in part by NSF Grants DMS-1301888 and DMS-1565352, a Simons Fellowship (#304043), and the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics at the University of Washington. This work started while enjoying the hospitality of the Institute for Advanced Study (Princeton) supported by The Wolfensohn Fund.
The second author was supported in part by NSF Grant DMS-1502236.
Article copyright:
© Copyright 2016
American Mathematical Society