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Mosc. Math. J., 2002, Volume 2, Number 3, Pages 435–475 (Mi mmj60)  

This article is cited in 14 scientific papers (total in 14 papers)

Toric residues and mirror symmetry

V. V. Batyrev, E. N. Materov

Eberhard Karls Universität Tübingen

Abstract: We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture, which has close relations to toric mirror symmetry. Our conjecture, we call it the toric residue mirror conjecture, is that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidence for that. The proposed conjecture suggests a simple method for computing Yukawa couplings for toric mirror Calabi–Yau hypersurfaces without solving systems of differential equations. We make several explicit computations for Calabi–Yau hypersurfaces in weighted projective spaces and in products of projective spaces.

Key words and phrases: Residues, toric varieties, intersection numbers, mirror symmetry.


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MSC: 14M25
Received: March 21, 2002

Citation: V. V. Batyrev, E. N. Materov, “Toric residues and mirror symmetry”, Mosc. Math. J., 2:3 (2002), 435–475

Citation in format AMSBIB
\by V.~V.~Batyrev, E.~N.~Materov
\paper Toric residues and mirror symmetry
\jour Mosc. Math.~J.
\yr 2002
\vol 2
\issue 3
\pages 435--475

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    This publication is cited in the following articles:
    1. Szenes A., Vergne M., “Toric reduction and a conjecture of Batyrev and Materov”, Invent. Math., 158:3 (2004), 453–495  crossref  mathscinet  zmath  adsnasa  isi
    2. Karu K., “Toric residue mirror conjecture for Calabi-Yau complete intersections”, J. Algebraic Geom., 14:4 (2005), 741–760  crossref  mathscinet  zmath  isi
    3. Cox D., Dickenstein A., “Codimension theorems for complete toric varieties”, Proc. Amer. Math. Soc., 133:11 (2005), 3153–3162  crossref  mathscinet  zmath  isi
    4. Khetan A., Soprounov I., “Combinatorial construction of toric residues”, Ann. Inst. Fourier (Grenoble), 55:2 (2005), 511–548  crossref  mathscinet  zmath  isi
    5. Borisov L.A., “Higher Stanley-Reisner rings and toric residues”, Compos. Math., 141:1 (2005), 161–174  crossref  mathscinet  zmath  isi
    6. Soprounov I., “Toric residue and combinatorial degree”, Trans. Amer. Math. Soc., 357:5 (2005), 1963–1975  crossref  mathscinet  zmath  isi
    7. Szenes A., Vergne M., “Mixed toric residues and tropical degenerations”, Topology, 45:3 (2006), 567–599  crossref  mathscinet  zmath  isi
    8. Curran R., Cattani E., “Restriction of $A$-discriminants and dual defect toric varieties”, J. Symbolic Comput., 42:1–2 (2007), 115–135  crossref  mathscinet  zmath  isi
    9. Jinzenji M., “Virtual structure constants as intersection numbers of moduli space of polynomial maps with two marked points”, Lett. Math. Phys., 86:2-3 (2008), 99–114  crossref  mathscinet  zmath  adsnasa  isi
    10. Melnikov I.V., Plesser M.R., “A (0,2) mirror map”, Journal of High Energy Physics, 2011, no. 2, 001  crossref  mathscinet  zmath  isi
    11. Trenner T., Wilson P.M.H., “Asymptotic Curvature of Moduli Spaces for Calabi-Yau Threefolds”, J Geom Anal, 21:2 (2011), 409–428  crossref  mathscinet  zmath  isi
    12. Kreuzer M., McOrist J., Melnikov I.V., Plesser M.R., “(0,2) deformations of linear sigma models”, Journal of High Energy Physics, 2011, no. 7, 044  crossref  mathscinet  zmath  isi
    13. Ilarion Melnikov, Savdeep Sethi, Eric Sharpe, “Recent developments in (0,2) mirror symmetry”, SIGMA, 8 (2012), 068, 28 pp.  mathnet  crossref  mathscinet
    14. Honma Y., Manabe M., “Local B-Model Yukawa Couplings From a-Twisted Correlators”, Prog. Theor. Exp. Phys., 2018, no. 7, 073A03  crossref  mathscinet  isi  scopus
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