Mirror symmetry and projective geometry of Reye congruences I
Authors:
Shinobu Hosono and Hiromichi Takagi
Journal:
J. Algebraic Geom. 23 (2014), 279-312
DOI:
https://doi.org/10.1090/S1056-3911-2013-00618-9
Published electronically:
July 15, 2013
MathSciNet review:
3166392
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Abstract |
References |
Additional Information
Abstract: Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in $\mathbb {P}^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in $\mathbb {P}^4$ branched over a curve. We also calculate BPS numbers of both $X$ and $Y$ (and also a related Calabi-Yau complete intersection $\tilde X_0$) using mirror symmetry.
References
- Paul S. Aspinwall, Brian R. Greene, and David R. Morrison, Multiple mirror manifolds and topology change in string theory, Phys. Lett. B 303 (1993), no. 3-4, 249–259. MR 1214080, DOI https://doi.org/10.1016/0370-2693%2893%2991428-P
- Victor V. Batyrev and Lev A. Borisov, On Calabi-Yau complete intersections in toric varieties, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 39–65. MR 1463173
- Victor V. Batyrev and Duco van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Comm. Math. Phys. 168 (1995), no. 3, 493–533. MR 1328251
- Lev Borisov and Andrei Căldăraru, The Pfaffian-Grassmannian derived equivalence, J. Algebraic Geom. 18 (2009), no. 2, 201–222. MR 2475813, DOI https://doi.org/10.1090/S1056-3911-08-00496-7
- Victor V. Batyrev and David A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J. 75 (1994), no. 2, 293–338. MR 1290195, DOI https://doi.org/10.1215/S0012-7094-94-07509-1
- Arnaud Beauville, L’application canonique pour les surfaces de type général, Invent. Math. 55 (1979), no. 2, 121–140 (French). MR 553705, DOI https://doi.org/10.1007/BF01390086
- M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys. 165 (1994), no. 2, 311–427. MR 1301851
- Tom Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), no. 3, 613–632. MR 1893007, DOI https://doi.org/10.1007/s002220100185
- Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21–74. MR 1115626, DOI https://doi.org/10.1016/0550-3213%2891%2990292-6
- F. Catanese, Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications, Invent. Math. 63 (1981), no. 3, 433–465. MR 620679, DOI https://doi.org/10.1007/BF01389064
- Pierre-Emmanuel Chaput, Scorza varieties and Jordan algebras, Indag. Math. (N.S.) 14 (2003), no. 2, 169–182. MR 2026812, DOI https://doi.org/10.1016/S0019-3577%2803%2990002-4
- François R. Cossec, Reye congruences, Trans. Amer. Math. Soc. 280 (1983), no. 2, 737–751. MR 716848, DOI https://doi.org/10.1090/S0002-9947-1983-0716848-4
- François R. Cossec and Igor V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 986969
- Igor Dolgachev and Vassil Kanev, Polar covariants of plane cubics and quartics, Adv. Math. 98 (1993), no. 2, 216–301. MR 1213725, DOI https://doi.org/10.1006/aima.1993.1016
- Christian van Enckevort and Duco van Straten, Monodromy calculations of fourth order equations of Calabi-Yau type, Mirror symmetry. V, AMS/IP Stud. Adv. Math., vol. 38, Amer. Math. Soc., Providence, RI, 2006, pp. 539–559. MR 2282974
- I. M. Gel′fand, A. V. Zelevinskiĭ, and M. M. Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 12–26 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 2, 94–106. MR 1011353, DOI https://doi.org/10.1007/BF01078777
- I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417
- R. Gopakumar and C. Vafa, M-Theory and Topological Strings–II, hep-th/9812127.
- S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nuclear Phys. B 433 (1995), no. 3, 501–552. MR 1319280, DOI https://doi.org/10.1016/0550-3213%2894%2900440-P
- Shinobu Hosono and Yukiko Konishi, Higher genus Gromov-Witten invariants of the Grassmannian, and the Pfaffian Calabi-Yau 3-folds, Adv. Theor. Math. Phys. 13 (2009), no. 2, 463–495. MR 2481271
- S. Hosono and H. Takagi, Duality between $S^2\mathbb {P}^4$ and the Double Quintic Symmetroid, preprint arXiv:1302.5881.
- S. Hosono and H. Takagi, Double Quintic Symmetroids, Reye Congruences, and their Derived Equivalence, preprint arXiv:1302.5883.
- Joe Harris and Loring W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), no. 1, 71–84. MR 721453, DOI https://doi.org/10.1016/0040-9383%2884%2990026-0
- D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2244106
- C. Ingalls and A. Kuznetsov, On nodal Enriques surfaces and quadric double solids, math. AG/arXive:1012. 3530.
- Sheldon Katz, Albrecht Klemm, and Cumrun Vafa, M-theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999), no. 5, 1445–1537. MR 1796683, DOI https://doi.org/10.4310/ATMP.1999.v3.n5.a6
- János Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36. MR 986434, DOI https://doi.org/10.1017/S0027763000001240
- A. Kuznetsov, Homological projective duality for Grassmannians of lines, arXiv:math/0610957.
- Alexander Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), no. 5, 1340–1369. MR 2419925, DOI https://doi.org/10.1016/j.aim.2008.03.007
- A. Kuznetsov, Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category, math. AG/arXiv:1011. 4146.
- Maplesoft, Waterloo Maple Inc. 2010.
- David R. Morrison, Compactifications of moduli spaces inspired by mirror symmetry, Astérisque 218 (1993), 243–271. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265317
- Shigeru Mukai, Duality of polarized $K3$ surfaces, New trends in algebraic geometry (Warwick, 1996) London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 311–326. MR 1714828, DOI https://doi.org/10.1017/CBO9780511721540.012
- Shigeru Mukai, Curves and Grassmannians, Algebraic geometry and related topics (Inchon, 1992) Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993, pp. 19–40. MR 1285374
- Shigeru Mukai, Polarized $K3$ surfaces of genus $18$ and $20$, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 264–276. MR 1201388, DOI https://doi.org/10.1017/CBO9780511662652.019
- Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
- Cristina Oliva, Algebraic cycles and Hodge theory on generalized Reye congruences, Compositio Math. 92 (1994), no. 1, 1–22. MR 1275718
- R. Pandharipande and R. P. Thomas, Stable pairs and BPS invariants, J. Amer. Math. Soc. 23 (2010), no. 1, 267–297. MR 2552254, DOI https://doi.org/10.1090/S0894-0347-09-00646-8
- PORTA, available at http://www. iwr. uni-heidelberg. de/groups/comopt/ software/PORTA/
- 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
- Einar Andreas Rødland, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian $G(2,7)$, Compositio Math. 122 (2000), no. 2, 135–149. MR 1775415, DOI https://doi.org/10.1023/A%3A1001847914402
- Edoardo Sernesi, Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 334, Springer-Verlag, Berlin, 2006. MR 2247603
- Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 382272, DOI https://doi.org/10.1007/BF01389674
- A. N. Tyurin, On intersections of quadrics, Russian Math. Surveys 30 (1975), 51–105.
- P. M. H. Wilson, The Kähler cone on Calabi-Yau threefolds, Invent. Math. 107 (1992), no. 3, 561–583. MR 1150602, DOI https://doi.org/10.1007/BF01231902
References
- P. S. Aspinwall, B. R. Greene, and D. R. Morrison, Multiple mirror manifolds and topology change in string theory, Phys. Lett. B 303 (1993), no. 3-4, 249–259. MR 1214080 (94g:32031), DOI https://doi.org/10.1016/0370-2693%2893%2991428-P
- V. V. Batyrev and L. A. Borisov, On Calabi-Yau complete intersections in toric varieties, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 39–65. MR 1463173 (98j:14052)
- V. V. Batyrev and D. van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Comm. Math. Phys. 168 (1995), no. 3, 493–533. MR 1328251 (96g:32037)
- L. Borisov and A. Căldăraru, The Pfaffian-Grassmannian derived equivalence, J. Algebraic Geom. 18 (2009), no. 2, 201–222. MR 2475813 (2010i:14069), DOI https://doi.org/10.1090/S1056-3911-08-00496-7
- V. V. Batyrev and D. A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J. 75 (1994), no. 2, 293–338. MR 1290195 (95j:14072), DOI https://doi.org/10.1215/S0012-7094-94-07509-1
- A. Beauville, L’application canonique pour les surfaces de type général, Invent. Math. 55 (1979), no. 2, 121–140 (French). MR 553705 (81m:14025), DOI https://doi.org/10.1007/BF01390086
- M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys. 165 (1994), no. 2, 311–427. MR 1301851 (95f:32029)
- T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), no. 3, 613–632. MR 1893007 (2003h:14027), DOI https://doi.org/10.1007/s002220100185
- P. Candelas, X. C. de la Ossa, P. S. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21–74. MR 1115626 (93b:32029), DOI https://doi.org/10.1016/0550-3213%2891%2990292-6
- F. Catanese, Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications, Invent. Math. 63 (1981), no. 3, 433–465. MR 620679 (83c:14026), DOI https://doi.org/10.1007/BF01389064
- P. E. Chaput, Scorza varieties and Jordan algebras, Indag. Math. (N.S.) 14 (2003), no. 2, 169–182. MR 2026812 (2005b:14084), DOI https://doi.org/10.1016/S0019-3577%2803%2990002-4
- F. R. Cossec, Reye congruences, Trans. Amer. Math. Soc. 280 (1983), no. 2, 737–751. MR 716848 (85b:14049), DOI https://doi.org/10.2307/1999644
- F. R. Cossec and I. V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston Inc., Boston, MA, 1989. MR 986969 (90h:14052)
- I. Dolgachev and V. Kanev, Polar covariants of plane cubics and quartics, Adv. Math. 98 (1993), no. 2, 216–301. MR 1213725 (94g:14029), DOI https://doi.org/10.1006/aima.1993.1016
- C. van Enckevort and D. van Straten, Monodromy calculations of fourth order equations of Calabi-Yau type, Mirror symmetry. V, AMS/IP Stud. Adv. Math., vol. 38, Amer. Math. Soc., Providence, RI, 2006, pp. 539–559. MR 2282974 (2007m:14057)
- I. M. Gel′fand, A. V. Zelevinskiĭ, and M. M. Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 12–26 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 2, 94–106. MR 1011353 (90m:22025), DOI https://doi.org/10.1007/BF01078777
- I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1994. MR 1264417 (95e:14045)
- R. Gopakumar and C. Vafa, M-Theory and Topological Strings–II, hep-th/9812127.
- S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nuclear Phys. B 433 (1995), no. 3, 501–552. MR 1319280 (96d:32028), DOI https://doi.org/10.1016/0550-3213%2894%2900440-P
- S. Hosono and Y. Konishi, Higher genus Gromov-Witten invariants of the Grassmannian, and the Pfaffian Calabi-Yau 3-folds, Adv. Theor. Math. Phys. 13 (2009), no. 2, 463–495. MR 2481271 (2009k:53231)
- S. Hosono and H. Takagi, Duality between $S^2\mathbb {P}^4$ and the Double Quintic Symmetroid, preprint arXiv:1302.5881.
- S. Hosono and H. Takagi, Double Quintic Symmetroids, Reye Congruences, and their Derived Equivalence, preprint arXiv:1302.5883.
- J. Harris and L. W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), no. 1, 71–84. MR 721453 (85c:14032), DOI https://doi.org/10.1016/0040-9383%2884%2990026-0
- D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2006. MR 2244106 (2007f:14013)
- C. Ingalls and A. Kuznetsov, On nodal Enriques surfaces and quadric double solids, math. AG/arXive:1012. 3530.
- S. Katz, A. Klemm, and C. Vafa, M-theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999), no. 5, 1445–1537. MR 1796683 (2002b:81124)
- J. Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36. MR 986434 (90e:14011)
- A. Kuznetsov, Homological projective duality for Grassmannians of lines, arXiv:math/0610957.
- A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), no. 5, 1340–1369. MR 2419925 (2009g:14019), DOI https://doi.org/10.1016/%5Cspace%20j.aim.2008.03.007
- A. Kuznetsov, Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category, math. AG/arXiv:1011. 4146.
- Maplesoft, Waterloo Maple Inc. 2010.
- D. R. Morrison, Compactifications of moduli spaces inspired by mirror symmetry, Astérisque 218 (1993), 243–271. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265317 (95d:32021)
- S. Mukai, Duality of polarized $K3$ surfaces, New trends in algebraic geometry (Warwick, 1996) London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 311–326. MR 1714828 (2000i:14057), DOI https://doi.org/10.1017/CBO9780511721540.012
- S. Mukai, Curves and Grassmannians, Algebraic geometry and related topics (Inchon, 1992) Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993, pp. 19–40. MR 1285374 (95i:14032)
- S. Mukai, Polarized $K3$ surfaces of genus $18$ and $20$, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 264–276. MR 1201388 (94a:14039), DOI https://doi.org/10.1017/CBO9780511662652.019
- T. Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894 (88m:14038)
- C. Oliva, Algebraic cycles and Hodge theory on generalized Reye congruences, Compositio Math. 92 (1994), no. 1, 1–22. MR 1275718 (95f:14016)
- R. Pandharipande and R. P. Thomas, Stable pairs and BPS invariants, J. Amer. Math. Soc. 23 (2010), no. 1, 267–297. MR 2552254 (2010i:14104), DOI https://doi.org/10.1090/S0894-0347-09-00646-8
- PORTA, available at http://www. iwr. uni-heidelberg. de/groups/comopt/ software/PORTA/
- M. 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 (89b:14016)
- E. A. Rødland, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian $G(2,7)$, Compositio Math. 122 (2000), no. 2, 135–149. MR 1775415 (2001h:14051), DOI https://doi.org/10.1023/A%3A1001847914402
- E. Sernesi, Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 334, Springer-Verlag, Berlin, 2006. MR 2247603 (2008e:14011)
- W. Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 0382272 (52 \#3157)
- A. N. Tyurin, On intersections of quadrics, Russian Math. Surveys 30 (1975), 51–105.
- P. M. H. Wilson, The Kähler cone on Calabi-Yau threefolds, Invent. Math. 107 (1992), no. 3, 561–583. MR 1150602 (93a:14037), DOI https://doi.org/10.1007/BF01231902
Additional Information
Shinobu Hosono
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba Meguro-ku, Tokyo 153-8914, Japan
Email:
hosono@ms.u-tokyo.ac.jp
Hiromichi Takagi
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba Meguro-ku, Tokyo 153-8914, Japan
Email:
takagi@ms.u-tokyo.ac.jp
Received by editor(s):
February 7, 2011
Received by editor(s) in revised form:
June 4, 2012
Published electronically:
July 15, 2013
Additional Notes:
The first author was supported in part by Grant-in-Aid for Scientific Research, C 18540014. The second author was supported in part by Grant-in-Aid for Young Scientists, B 20740005
Article copyright:
© Copyright 2013
University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.