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

Buckley, Anita; Szendrői, Balázs

doi:10.1090/S1056-3911-05-00403-0

Authors: Anita Buckley and Balázs Szendrői
Journal: J. Algebraic Geom. 14 (2005), 601-622
DOI: https://doi.org/10.1090/S1056-3911-05-00403-0
Published electronically: April 27, 2005
MathSciNet review: 2147356
Full-text PDF

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.

altinok S. Altınok, Graded rings corresponding to polarised K3 surfaces and $\mathbb {Q}$-Fano $3$-folds, Ph.D. Thesis, University of Warwick, 1998.

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
M. F. Atiyah and G. B. Segal, The index of elliptic operators. II, Ann. of Math. (2) 87 (1968), 531–545. MR 236951, DOI https://doi.org/10.2307/1970716
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI https://doi.org/10.2307/1970717
Victor V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), no. 3, 493–535. MR 1269718

bb L. A. Borisov, Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties, alg-geom/9310001.

Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI https://doi.org/10.1006/jsco.1996.0125

anita A. Buckley, Orbifold Riemann–Roch for threefolds and applications to Calabi–Yaus, Ph.D. Thesis, University of Warwick, 2003.

P. Candelas, A. M. Dale, C. A. Lütken, and R. C. Schimmrigk, Complete intersection Calabi-Yau manifolds, Frontiers of high energy physics (London, 1986) Hilger, Bristol, 1987, pp. 88–134. MR 923602
P. Candelas, M. Lynker, and R. Schimmrigk, Calabi-Yau manifolds in weighted ${\bf P}_4$, Nuclear Phys. B 341 (1990), no. 2, 383–402. MR 1067295, DOI https://doi.org/10.1016/0550-3213%2890%2990185-G
Alessio Corti and Miles Reid, Weighted Grassmannians, Algebraic geometry, de Gruyter, Berlin, 2002, pp. 141–163. MR 1954062
A. R. Fletcher, Contributions to Riemann-Roch on projective $3$-folds with only canonical singularities and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 221–231. MR 927958
A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 101–173. MR 1798982
Kenneth Intriligator, David R. Morrison, and Nathan Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nuclear Phys. B 497 (1997), no. 1-2, 56–100. MR 1467883, DOI https://doi.org/10.1016/S0550-3213%2897%2900279-4
Tetsuro Kawasaki, The Riemann-Roch theorem for complex $V$-manifolds, Osaka Math. J. 16 (1979), no. 1, 151–159. MR 527023
Maximilian Kreuzer and Harald Skarke, Complete classification of reflexive polyhedra in four dimensions, Adv. Theor. Math. Phys. 4 (2000), no. 6, 1209–1230. MR 1894855, DOI https://doi.org/10.4310/ATMP.2000.v4.n6.a2
Maximilian Kreuzer, Erwin Riegler, and David A. Sahakyan, Toric complete intersections and weighted projective space, J. Geom. Phys. 46 (2003), no. 2, 159–173. MR 1967149, DOI https://doi.org/10.1016/S0393-0440%2802%2900124-9
Stavros Argyrios Papadakis and Miles Reid, Kustin-Miller unprojection without complexes, J. Algebraic Geom. 13 (2004), no. 3, 563–577. MR 2047681, DOI https://doi.org/10.1090/S1056-3911-04-00343-1
Miles Reid, Canonical $3$-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 273–310. MR 605348
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
Balázs Szendrői, Calabi-Yau threefolds with a curve of singularities and counterexamples to the Torelli problem. II, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 2, 193–204. MR 1765909, DOI https://doi.org/10.1017/S0305004100004588
B. Toen, Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford, $K$-Theory 18 (1999), no. 1, 33–76 (French, with English and French summaries). MR 1710187, DOI https://doi.org/10.1023/A%3A1007791200714

Additional Information

Anita Buckley
Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, Ljubljana 1000, Slovenia
Email: anita.buckley@fmf.uni-lj.si

Balázs Szendrői
Affiliation: Department of Mathematics, Utrecht University, PO. Box 80010, NL-3508 TA Utrecht, The Netherlands
Email: szendroi@math.uu.nl

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.