Toric residue mirror conjecture for Calabi-Yau complete intersections
Author:
Kalle Karu
Journal:
J. Algebraic Geom. 14 (2005), 741-760
DOI:
https://doi.org/10.1090/S1056-3911-05-00410-8
Published electronically:
April 27, 2005
MathSciNet review:
2147350
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Abstract |
References |
Additional Information
Abstract: We prove two conjectures of V. Batyrev and E. Materov. The first one is the toric residue mirror conjecture for Calabi-Yau complete intersections in Gorenstein toric Fano varieties. The second conjecture relates the homogeneous parts of the toric residue to the mixed volumes of polytopes.
- Gottfried Barthel, Jean-Paul Brasselet, Karl-Heinz Fieseler, and Ludger Kaup, Equivariant intersection cohomology of toric varieties, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998) Contemp. Math., vol. 241, Amer. Math. Soc., Providence, RI, 1999, pp. 45–68. MR 1718136, DOI https://doi.org/10.1090/conm/241/03627
- Victor V. Batyrev and Evgeny N. Materov, Toric residues and mirror symmetry, Mosc. Math. J. 2 (2002), no. 3, 435–475. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1988969, DOI https://doi.org/10.17323/1609-4514-2002-2-3-435-475
- Victor V. Batyrev and Evgeny N. Materov, Mixed toric residues and Calabi-Yau complete intersections, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001) Fields Inst. Commun., vol. 38, Amer. Math. Soc., Providence, RI, 2003, pp. 3–26. MR 2019144, DOI https://doi.org/10.17323/1609-4514-2002-2-3-435-475
B1 L. A. Borisov, Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties, preprint arXiv:math.AG/9310001.
- Lev A. Borisov, Higher-Stanley-Reisner rings and toric residues, Compos. Math. 141 (2005), no. 1, 161–174. MR 2099774, DOI https://doi.org/10.1112/S0010437X04000831
- Michel Brion, The structure of the polytope algebra, Tohoku Math. J. (2) 49 (1997), no. 1, 1–32. MR 1431267, DOI https://doi.org/10.2748/tmj/1178225183
- Michel Brion and Michèle Vergne, Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 5, 715–741 (English, with English and French summaries). MR 1710758, DOI https://doi.org/10.1016/S0012-9593%2801%2980005-7
- Eduardo Cattani, Alicia Dickenstein, and Bernd Sturmfels, Residues and resultants, J. Math. Sci. Univ. Tokyo 5 (1998), no. 1, 119–148. MR 1617074
- David A. Cox, Toric residues, Ark. Mat. 34 (1996), no. 1, 73–96. MR 1396624, DOI https://doi.org/10.1007/BF02559508
- András Szenes and Michèle Vergne, Toric reduction and a conjecture of Batyrev and Materov, Invent. Math. 158 (2004), no. 3, 453–495. MR 2104791, DOI https://doi.org/10.1007/s00222-004-0375-2
BBFK G. Barthel, J.-P. Brasselet, K.-H. Fieseler, L. Kaup, Equivariant intersection cohomology of toric varieties, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 45–68, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, 1999.
BM1 V. V. Batyrev, E. N. Materov, Toric Residues and Mirror Symmetry, Dedicated to Yuri I. Manin on the occasion of his 65th birthday. Mosc. Math. J. 2 (2002), no. 3, 435–475.
BM2 V. V. Batyrev, E. N. Materov, Mixed toric residues and Calabi-Yau complete intersections, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001), 3–26, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003.
B1 L. A. Borisov, Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties, preprint arXiv:math.AG/9310001.
B2 L. A. Borisov, Higher Stanley-Reisner rings and toric residues, Compos. Math. 141 (2005), no. 1, 161–174.
B M. Brion, The structure of the polytope algebra, Tohoku Math. J. (2) 49 (1997), no. 1, 1–32.
BV M. Brion, M. Vergne, Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue, Ann. Sci. Ãcole Norm. Sup. (4) 32 (1999), no. 5, 715–741.
CDS E. Cattani, A. Dickenstein, B. Sturmfels, Residues and resultants, J. Math. Sci. Univ. Tokyo 5 (1998), no. 1, 119–148 .
C D. Cox, Toric residues, Ark. Mat. 34 (1996), no. 1, 73–96.
SV A. Szenes, M. Vergne, Toric reduction and a conjecture of Batyrev and Materov, Invent. Math. 158 (2004), no. 3, 453–495.
Additional Information
Kalle Karu
Affiliation:
Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, B.C. Canada V6T 1Z2
Email:
karu@math.ubc.ca
Received by editor(s):
October 20, 2004
Published electronically:
April 27, 2005
