Towards the second main theorem on complements

Prokhorov, Yu.; Shokurov, V.

doi:10.1090/S1056-3911-08-00498-0

Authors: Yu. G. Prokhorov and V. V. Shokurov
Journal: J. Algebraic Geom. 18 (2009), 151-199
DOI: https://doi.org/10.1090/S1056-3911-08-00498-0
Published electronically: April 24, 2008
MathSciNet review: 2448282
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Abstract | References | Additional Information

Abstract: We prove the boundedness of complements modulo two conjectures: Borisov–Alexeev conjecture and effective adjunction for fibre spaces. We discuss the last conjecture and prove it in two particular cases.

References

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

Florin Ambro, The adjunction conjecture and its applications, PhD thesis, The Johns Hopkins University, http://arXiv.org:math/9903060, 1999.

Florin Ambro, Shokurov’s boundary property, J. Differential Geom. 67 (2004), no. 2, 229–255. MR 2153078

Florin Ambro, The moduli $b$-divisor of an lc-trivial fibration, Compos. Math. 141 (2005), no. 2, 385–403. MR 2134273, DOI https://doi.org/10.1112/S0010437X04001071

A. A. Borisov and L. A. Borisov, Singular toric Fano three-folds, Mat. Sb. 183 (1992), no. 2, 134–141 (Russian); English transl., Russian Acad. Sci. Sb. Math. 75 (1993), no. 1, 277–283. MR 1166957, DOI https://doi.org/10.1070/SM1993v075n01ABEH003385

Raimund Blache, The structure of l.c. surfaces of Kodaira dimension zero. I, J. Algebraic Geom. 4 (1995), no. 1, 137–179. MR 1299007

Alexandr Borisov, Boundedness theorem for Fano log-threefolds, J. Algebraic Geom. 5 (1996), no. 1, 119–133. MR 1358037

Alexandr Borisov, Boundedness of Fano threefolds with log-terminal singularities of given index, J. Math. Sci. Univ. Tokyo 8 (2001), no. 2, 329–342. MR 1837167

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

J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR 0087708

Osamu Fujino and Shigefumi Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, 167–188. MR 1863025

Takao Fujita, Zariski decomposition and canonical rings of elliptic threefolds, J. Math. Soc. Japan 38 (1986), no. 1, 19–37. MR 816221, DOI https://doi.org/10.2969/jmsj/03810019

Osamu Fujino, Applications of Kawamata’s positivity theorem, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), no. 6, 75–79. MR 1712648

Osamu Fujino, Abundance theorem for semi log canonical threefolds, Duke Math. J. 102 (2000), no. 3, 513–532. MR 1756108, DOI https://doi.org/10.1215/S0012-7094-00-10237-2

Osamu Fujino, Higher direct images of log canonical divisors, J. Differential Geom. 66 (2004), no. 3, 453–479. MR 2106473

Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157

V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. MR 1668579

Shihoko Ishii, The global indices of log Calabi-Yau varieties – A supplement to Fujino’s paper: The indices of log canonical singularities –, math.AG/0003060, 2000.

V. A. Iskovskikh, $b$-divisors and Shokurov functional algebras, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 8–20 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 1(240) (2003), 4–15. MR 1993746

Yujiro Kawamata, Subadjunction of log canonical divisors for a subvariety of codimension $2$, Birational algebraic geometry (Baltimore, MD, 1996) Contemp. Math., vol. 207, Amer. Math. Soc., Providence, RI, 1997, pp. 79–88. MR 1462926, DOI https://doi.org/10.1090/conm/207/02721

Yujiro Kawamata, Subadjunction of log canonical divisors. II, Amer. J. Math. 120 (1998), no. 5, 893–899. MR 1646046

Sean Keel, Intersection theory of moduli space of stable $n$-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574. MR 1034665, DOI https://doi.org/10.1090/S0002-9947-1992-1034665-0

Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI https://doi.org/10.2969/aspm/01010283

Finn F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $M_{g,n}$, Math. Scand. 52 (1983), no. 2, 161–199. MR 702953, DOI https://doi.org/10.7146/math.scand.a-12001

K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. (2) 77 (1963), 563–626; ibid. 78 (1963), 1–40. MR 0184257, DOI https://doi.org/10.2307/1970500

Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991; Astérisque No. 211 (1992) (1992). MR 1225842

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 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

James McKernan, Boundedness of log terminal Fano pairs of bounded index, math.AG/0205214, 2002.

V. V. Nikulin, del Pezzo surfaces with log-terminal singularities. III, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1316–1334, 1338 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 657–675. MR 1039966, DOI https://doi.org/10.1070/IM1990v035n03ABEH000721

Yuri G. Prokhorov, Lectures on complements on log surfaces, MSJ Memoirs, vol. 10, Mathematical Society of Japan, Tokyo, 2001. MR 1830440

---, On Zariski decomposition problem, Proc. Steklov Inst. Math. 240 (2003), 37–65.

Yu. G. Prokhorov and V. V. Shokurov, The first fundamental theorem on complements: from global to local, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 6, 99–128 (Russian, with Russian summary); English transl., Izv. Math. 65 (2001), no. 6, 1169–1196. MR 1892905, DOI https://doi.org/10.1070/IM2001v065n06ABEH000366

V. G. Sarkisov, Birational automorphisms of conic bundles, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 918–945, 974 (Russian). MR 587343

V. V. Shokurov, 3-fold log flips. Appendix by Yujiro Kawamata: The minimal discrepancy coefficients of terminal singularities in dimension three, Russ. Acad. Sci., Izv., Math. 40 (1993), no. 1, 95–202.

V. V. Shokurov, Complements on surfaces, J. Math. Sci. (New York) 102 (2000), no. 2, 3876–3932. Algebraic geometry, 10. MR 1794169, DOI https://doi.org/10.1007/BF02984106

Qi Zhang, Rational connectedness of log ${\bf Q}$-Fano varieties, J. Reine Angew. Math. 590 (2006), 131–142. MR 2208131, DOI https://doi.org/10.1515/CRELLE.2006.006

Additional Information

Yu. G. Prokhorov
Affiliation: Department of Algebra, Faculty of Mathematics, Moscow State University, Moscow 117234, Russia
Email: prokhoro@mech.math.msu.su

V. V. Shokurov
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Address at time of publication: Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina str. 8, 119991, Moscow, Russia
Email: shokurov@math.jhu.edu

Received by editor(s): November 5, 2006
Received by editor(s) in revised form: September 16, 2007
Published electronically: April 24, 2008
Additional Notes: The first author was partially supported by grants CRDF-RUM, No. 1-2692-MO-05 and RFBR, No. 05-01-00353-a, 06-01-72017. The second author was partially supported by NSF grant DMS-0400832.