Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Orbifold Riemann-Roch for threefolds with an application to Calabi-Yau geometry

Authors: Anita Buckley and Balázs Szendroi
Journal: J. Algebraic Geom. 14 (2005), 601-622
Published electronically: April 27, 2005
MathSciNet review: 2147356
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Abstract | References | Additional Information

Abstract: We prove an orbifold Riemann-Roch formula for a polarized complex 3-fold $(X,D)$. As an application, we construct new families of projective Calabi-Yau threefolds.

References [Enhancements On Off] (What's this?)

  • 1. S. Altinok, Graded rings corresponding to polarised K3 surfaces and $\mathbb{Q} $-Fano $3$-folds, Ph.D. Thesis, University of Warwick, 1998.
  • 2. S. Altinok, G. Brown and M. Reid, Fano $3$-folds, $K3$surfaces and graded rings, in: Topology and geometry: commemorating SISTAG (ed. A. J. Berrick et al), 25-53, Contemp. Math., 314, Amer. Math. Soc., Providence, RI, 2002. MR 1941620 (2004c:14077)
  • 3. M. F. Atiyah and G. B. Segal, The index of elliptic operators. II, Ann. of Math. 87 (1968) 531-545. MR 0236951 (38:5244)
  • 4. M. F.Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. 87 (1968) 546-604. MR 0236952 (38:5245)
  • 5. V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994) 493-535. MR 1269718 (95c:14046)
  • 6. L. A. Borisov, Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties, alg-geom/9310001.
  • 7. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp. 24 (1997) 235-265. See also http://www.maths. MR 1484478
  • 8. A. Buckley, Orbifold Riemann-Roch for threefolds and applications to Calabi-Yaus, Ph.D. Thesis, University of Warwick, 2003.
  • 9. P. Candelas, A. M. Dale, C. A. Lütken and R. C. Schimmrigk, Complete intersection Calabi-Yau manifolds, Frontiers of high energy physics (London, 1986, ed. I. G. Halliday), 88-134, Hilger, Bristol, 1987. MR 0923602 (89e:14040)
  • 10. P. Candelas, M. Lynker and R. Schimmrigk, Calabi-Yau manifolds in weighted ${\mathbb P}^4$, Nuclear Phys. B 341 (1990) 383-402. MR 1067295 (91m:14062)
  • 11. A. Corti and M. Reid, Weighted Grassmannians, Algebraic geometry. A volume in memory of Paolo Francia (ed. M. C. Beltrametti et al), 141-163, de Gruyter, Berlin, 2002. MR 1954062 (2003m:14076)
  • 12. A. R. Fletcher, Contributions to Riemann-Roch on projective $3$-folds with only canonical singularities and applications Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985, ed. S. J. Bloch), 221-231, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987. MR 0927958 (89h:14032)
  • 13. A. R. Fletcher, Working with weighted complete intersections, Explicit birational geometry of $3$-folds (ed. A. Corti and M. Reid), 101-173, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000. MR 1798982 (2001k:14089)
  • 14. K. Intriligator, D. R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nuclear Phys. B 497 (1997) 56-100. MR 1467883 (98h:81117)
  • 15. Kawasaki, T. The Riemann-Roch theorem for complex $V$-manifolds, Osaka J. Math. 16 (1979) 151-159. MR 0527023 (80f:58042)
  • 16. M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four dimensions, Adv. Theor. Math. Phys. 4 (2002) 1209-1230. MR 1894855 (2003e:52020)
  • 17. M. Kreuzer, E. Riegler and D. A. Sahakyan, Toric complete intersections and weighted projective space, J. Geom. Phys. 46 (2003) 159-173. MR 1967149 (2004h:14044)
  • 18. S. Papadakis and M. Reid, Kustin-Miller unprojection without complexes, J. Algebraic Geom. 13 (2004) 563-577. MR 2047681
  • 19. M. Reid, Canonical $3$-folds, Journées de géométrie algébrique d'Angers (ed. A. Beauville), 1979, 273-310. MR 0605348 (82i:14025)
  • 20. M. Reid, Minimal models of canonical 3-folds, Algebraic Varieties and Analytic Varieties (ed. S. Iitaka), Advanced Studies in Pure Mathematics 1 (1983) 131-180. MR 0715649 (86a:14010)
  • 21. M. Reid, Young person's guide to canonical singularities, Algebraic geometry (Bowdoin 1985, ed. S. Bloch), 345-414, Proc. Sympos. Pure Math. 46, Part 1, AMS, Providence, RI, 1987. MR 0927963 (89b:14016)
  • 22. B. Szendroi, Calabi-Yau threefolds with a curve of singularities and counterexamples to the Torelli problem II, Math. Proc. Cambridge Philos. Soc. 129 (2000) 193-204. MR 1765909 (2002a:14046)
  • 23. B. Toen, Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford, $K$-Theory 18 (1999) 33-76. MR 1710187 (2000h:14010)

Additional Information

Anita Buckley
Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, Ljubljana 1000, Slovenia

Balázs Szendroi
Affiliation: Department of Mathematics, Utrecht University, PO. Box 80010, NL-3508 TA Utrecht, The Netherlands

Received by editor(s): September 9, 2003
Received by editor(s) in revised form: June 13, 2004
Published electronically: April 27, 2005
Additional Notes: We thank the Isaac Newton Institute, Cambridge for hospitality while part of this research was conducted, and the Mathematics Institute of the University of Warwick for supporting the research of A. B. with a Special Research Studentship.

American Mathematical Society