Abstract
For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kähler moduli space.
For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum theory and all the quantum corrections attached to the extremal ray vanish.
-
[vB] V. V. Batyrev, "Birational Calabi-Yau $n$-folds have equal Betti numbers," in New Trends in Algebraic Geometry, Cambridge: Cambridge Univ. Press, 1999, vol. 264, pp. 1-11.
@incollection {vB, MRKEY = {1714818},
AUTHOR = {Batyrev, Victor V.},
TITLE = {Birational {C}alabi-{Y}au {$n$}-folds have equal {B}etti numbers},
BOOKTITLE = {New Trends in Algebraic Geometry},
VENUE={{W}arwick, 1996},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {264},
PAGES = {1--11},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1999},
MRCLASS = {14J32 (14E30 14G40)},
MRNUMBER = {2000i:14059},
MRREVIEWER = {Mark Gross},
ZBLNUMBER = {0955.14028},
} -
[BK]
A. Bertram and H. P. Kley, "New recursions for genus-zero Gromov-Witten invariants," Topology, vol. 44, iss. 1, pp. 1-24, 2005.@article {BK, MRKEY = {2103998},
AUTHOR = {Bertram, Aaron and Kley, Holger P.},
TITLE = {New recursions for genus-zero {G}romov-{W}itten invariants},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {44},
YEAR = {2005},
NUMBER = {1},
PAGES = {1--24},
ISSN = {0040-9383},
CODEN = {TPLGAF},
MRCLASS = {14N35},
MRNUMBER = {2005h:14131},
MRREVIEWER = {Dragos Nicolae Oprea},
DOI = {10.1016/j.top.2001.12.002},
ZBLNUMBER = {1083.14064},
} -
[BG] J. Bryan and T. Graber, "The crepant resolution conjecture," in Algebraic Geometry—Seattle 2005. Part 1, Providence, RI: Amer. Math. Soc., 2009, vol. 80, pp. 23-42.
@incollection {BG, MRKEY = {2483931},
AUTHOR = {Bryan, Jim and Graber, Tom},
TITLE = {The crepant resolution conjecture},
BOOKTITLE = {Algebraic Geometry---{S}eattle 2005. {P}art 1},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME= {80},
PAGES = {23--42},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2009},
MRCLASS = {14N35 (14E15)},
MRNUMBER = {2009m:14083},
MRREVIEWER = {Hsian-Hua Tseng},
ZBLNUMBER = {05548145},
} -
[Cornalba] M. Cornalba, "Two theorems on modifications of analytic spaces," Invent. Math., vol. 20, pp. 227-247, 1973.
@article {Cornalba, MRKEY = {0481098},
AUTHOR = {Cornalba, Maurizio},
TITLE = {Two theorems on modifications of analytic spaces},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {20},
YEAR = {1973},
PAGES = {227--247},
ISSN = {0020-9910},
MRCLASS = {32C45},
MRNUMBER = {58 \#1244},
DOI = {10.1007/BF01394096},
ZBLNUMBER = {0264.32006},
} -
[CoKa] D. A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Providence, RI: Amer. Math. Soc., 1999.
@book {CoKa, MRKEY = {1677117},
AUTHOR = {Cox, David A. and Katz, Sheldon},
TITLE = {Mirror Symmetry and Algebraic Geometry},
SERIES = {Mathematical Surveys and Monographs},
NUMBER = {68},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1999},
PAGES = {xxii+469},
ISBN = {0-8218-1059-6},
MRCLASS = {14J32 (14-02 14M25 14N10 14N35 32G81 32J81 32Q25)},
MRNUMBER = {2000d:14048},
MRREVIEWER = {Andreas Gathmann},
ZBLNUMBER = {0951.14026},
} -
[Fulton] W. Fulton, Intersection Theory, New York: Springer-Verlag, 1984, vol. 2.
@book {Fulton, MRKEY = {732620},
AUTHOR = {Fulton, William},
TITLE = {Intersection Theory},
SERIES = {Ergeb. Math. Grenzgeb.},
VOLUME = {2},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1984},
PAGES = {xi+470},
ISBN = {3-540-12176-5},
MRCLASS = {14C17 (14-02 14C40)},
MRNUMBER = {85k:14004},
MRREVIEWER = {Werner Kleinert},
ZBLNUMBER = {0541.14005},
} -
[Givental] A. Givental, "A mirror theorem for toric complete intersections," in Topological Field Theory, Primitive Forms and Related Topics, Boston, MA: Birkhäuser, 1998, vol. 160, pp. 141-175.
@incollection {Givental, MRKEY = {1653024},
AUTHOR = {Givental, Alexander},
TITLE = {A mirror theorem for toric complete intersections},
BOOKTITLE = {Topological Field Theory, Primitive Forms and Related Topics},
VENUE={{K}yoto, 1996},
SERIES = {Progr. Math.},
VOLUME = {160},
PAGES = {141--175},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1998},
MRCLASS = {14M25 (14J32 14N35)},
MRNUMBER = {2000a:14063},
MRREVIEWER = {Charles F. Doran},
ZBLNUMBER = {0936.14031},
} -
[HuZh] J. Hu and W. Zhang, "Mukai flop and Ruan cohomology," Math. Ann., vol. 330, iss. 3, pp. 577-599, 2004.
@article {HuZh, MRKEY = {2099194},
AUTHOR = {Hu, Jianxun and Zhang, Wanchuan},
TITLE = {Mukai flop and {R}uan cohomology},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {330},
YEAR = {2004},
NUMBER = {3},
PAGES = {577--599},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {14N35 (53D45)},
MRNUMBER = {2005g:14105},
MRREVIEWER = {David Ploog},
DOI = {10.1007/s00208-004-0561-y},
ZBLNUMBER = {1078.14081},
} -
[Huyb1] D. Huybrechts, "Birational symplectic manifolds and their deformations," J. Differential Geom., vol. 45, iss. 3, pp. 488-513, 1997.
@article {Huyb1, MRKEY = {1472886},
AUTHOR = {Huybrechts, Daniel},
TITLE = {Birational symplectic manifolds and their deformations},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {45},
YEAR = {1997},
NUMBER = {3},
PAGES = {488--513},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {32G05 (32J18)},
MRNUMBER = {98e:32031},
MRREVIEWER = {Mark Gross},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214459840},
ZBLNUMBER = {0917.53010},
} -
[IoPa] E. Ionel and T. H. Parker, "The symplectic sum formula for Gromov-Witten invariants," Ann. of Math., vol. 159, iss. 3, pp. 935-1025, 2004.
@article {IoPa, MRKEY = {2113018},
AUTHOR = {Ionel, Eleny-Nicoleta and Parker, Thomas H.},
TITLE = {The symplectic sum formula for {G}romov-{W}itten invariants},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {159},
YEAR = {2004},
NUMBER = {3},
PAGES = {935--1025},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {53D45 (14N35 57R17)},
MRNUMBER = {2006b:53110},
MRREVIEWER = {Vicente Mu{ñ}oz},
DOI = {10.4007/annals.2004.159.935},
ZBLNUMBER = {1075.53092},
} -
[KoMo2] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge: Cambridge Univ. Press, 1998, vol. 134.
@book {KoMo2, MRKEY = {1658959},
AUTHOR = {Koll{á}r, J{á}nos and Mori, Shigefumi},
TITLE = {Birational Geometry of Algebraic Varieties},
SERIES = {Cambridge Tracts in Math.},
VOLUME = {134},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1998},
PAGES = {viii+254},
ISBN = {0-521-63277-3},
MRCLASS = {14E30},
MRNUMBER = {2000b:14018},
MRREVIEWER = {Mark Gross},
ZBLNUMBER = {0926.14003},
} -
[ypL] Y. -P. Lee, "Quantum Lefschetz hyperplane theorem," Invent. Math., vol. 145, iss. 1, pp. 121-149, 2001.
@article {ypL, MRKEY = {1839288},
AUTHOR = {Lee, Y.-P.},
TITLE = {Quantum {L}efschetz hyperplane theorem},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {145},
YEAR = {2001},
NUMBER = {1},
PAGES = {121--149},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14N35},
MRNUMBER = {2002i:14049},
MRREVIEWER = {Gilberto Bini},
DOI = {10.1007/s002220100145},
ZBLNUMBER = {1082.14056},
} -
[LP] Y. -P. Lee and R. Pandharipande, "A reconstruction theorem in quantum cohomology and quantum $K$-theory," Amer. J. Math., vol. 126, iss. 6, pp. 1367-1379, 2004.
@article {LP, MRKEY = {2102400},
AUTHOR = {Lee, Y.-P. and Pandharipande, R.},
TITLE = {A reconstruction theorem in quantum cohomology and quantum {$K$}-theory},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {126},
YEAR = {2004},
NUMBER = {6},
PAGES = {1367--1379},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {14N35 (19E08)},
MRNUMBER = {2006c:14082},
MRREVIEWER = {Andrew Kresch},
URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v126/126.6lee.pdf},
ZBLNUMBER = {1080.14065},
} -
[Li] J. Li, "A degeneration formula of GW-invariants," J. Differential Geom., vol. 60, iss. 2, pp. 199-293, 2002.
@article {Li, MRKEY = {1938113},
AUTHOR = {Li, Jun},
TITLE = {A degeneration formula of {GW}-invariants},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {60},
YEAR = {2002},
NUMBER = {2},
PAGES = {199--293},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {14N35},
MRNUMBER = {2004k:14096},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1090351102},
ZBLNUMBER = {1063.14069},
} -
[LiRu] A. Li and Y. Ruan, "Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3 folds," Invent. Math., vol. 145, iss. 1, pp. 151-218, 2001.
@article {LiRu, MRKEY = {1839289},
AUTHOR = {Li, An-Min and Ruan, Yongbin},
TITLE = {Symplectic surgery and {G}romov-{W}itten invariants of {C}alabi-{Y}au 3~folds},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {145},
YEAR = {2001},
NUMBER = {1},
PAGES = {151--218},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {53D45 (14J32 14N35 32Q25 32Q65)},
MRNUMBER = {2002g:53158},
MRREVIEWER = {Ignasi Mundet-Riera},
DOI = {10.1007/s002220100146},
ZBLNUMBER = {1062.53073},
} -
[LLY1] B. H. Lian, K. Liu, and S. Yau, "Mirror principle. I," Asian J. Math., vol. 1, iss. 4, pp. 729-763, 1997.
@article {LLY1, MRKEY = {1621573},
AUTHOR = {Lian, Bong H. and Liu, Kefeng and Yau, Shing-Tung},
TITLE = {Mirror principle. {I}},
JOURNAL = {Asian J. Math.},
FJOURNAL = {The Asian Journal of Mathematics},
VOLUME = {1},
YEAR = {1997},
NUMBER = {4},
PAGES = {729--763},
ISSN = {1093-6106},
MRCLASS = {14N10 (14D20 14J32)},
MRNUMBER = {99e:14062},
MRREVIEWER = {Andreas Gathmann},
ZBLNUMBER = {0953.14026},
} -
[LLY3] B. H. Lian, K. Liu, and S. Yau, "Mirror principle III," in Surv. Differ. Geom. 7, Somerville, MA: Internat. Press, 2000, pp. 433-474.
@incollection{LLY3,
author = {Lian, Bong H. and Liu, Kefeng and Yau, Shing-Tung},
TITLE = {Mirror principle III},
BOOKTITLE={Surv. Differ. Geom. {\bf 7}},
PUBLISHER={Internat. Press},
ADDRESS={Somerville, MA},
PAGES={433--474},
YEAR={2000},
MRNUMBER={1919433},
} -
[LiuYau] C. -H. Liu and S. -T. Yau, A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups.
@misc{LiuYau,
author={Liu, C.-H. and Yau, S.-T.},
TITLE={A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups},
ARXIV={math.AG/0408147},
} -
[Manin] J. I. Manin, "Correspondences, motifs and monoidal transformations," Mat. Sb., vol. 77 (119), pp. 475-507, 1968.
@article {Manin, MRKEY = {0258836},
AUTHOR = {Manin, Ju. I.},
TITLE = {Correspondences, motifs and monoidal transformations},
JOURNAL = {Mat. Sb.},
VOLUME = {77 (119)},
NOTE={in Russian; translated in {\it Math. {\rm USSR} Sb.} {\bf 6} (1968), 439--470},
YEAR = {1968},
PAGES = {475--507},
MRCLASS = {14.47},
MRNUMBER = {41 \#3482},
MRREVIEWER = {J. W. S. Cassels},
ZBLNUMBER = {0199.24803},
} -
[MP] D. Maulik and R. Pandharipande, "A topological view of Gromov-Witten theory," Topology, vol. 45, iss. 5, pp. 887-918, 2006.
@article {MP, MRKEY = {2248516},
AUTHOR = {Maulik, D. and Pandharipande, R.},
TITLE = {A topological view of {G}romov-{W}itten theory},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {45},
YEAR = {2006},
NUMBER = {5},
PAGES = {887--918},
ISSN = {0040-9383},
CODEN = {TPLGAF},
MRCLASS = {14N35 (53D45)},
MRNUMBER = {2007e:14092},
MRREVIEWER = {Renzo Cavalieri},
DOI = {10.1016/j.top.2006.06.002},
ZBLNUMBER = {1112.14065},
} -
[Morrison] D. R. Morrison, "Beyond the Kähler cone," in Proc. of the Hirzebruch 65 Conference on Algebraic Geometry, Ramat Gan, 1996, pp. 361-376.
@inproceedings {Morrison, MRKEY = {1360514},
AUTHOR = {Morrison, David R.},
TITLE = {Beyond the {K}ähler cone},
BOOKTITLE = {Proc. of the {H}irzebruch 65 {C}onference on {A}lgebraic {G}eometry},
VENUE={{R}amat {G}an, 1993},
SERIES = {Israel Math. Conf. Proc.},
VOLUME = {9},
PAGES = {361--376},
PUBLISHER = {Bar-Ilan Univ.},
ADDRESS = {Ramat Gan},
YEAR = {1996},
MRCLASS = {32G81 (14J32 14N10 32G20 32J81)},
MRNUMBER = {97a:32020},
MRREVIEWER = {Anatoly Libgober},
ZBLNUMBER={0847.32034},
} -
[Ruan] Y. Ruan, "Surgery, quantum cohomology and birational geometry," in Northern California Symplectic Geometry Seminar, Providence, RI: Amer. Math. Soc., 1999, vol. 196, pp. 183-198.
@incollection {Ruan, MRKEY = {1736218},
AUTHOR = {Ruan, Yongbin},
TITLE = {Surgery, quantum cohomology and birational geometry},
BOOKTITLE = {Northern {C}alifornia {S}ymplectic {G}eometry {S}eminar},
SERIES = {Amer. Math. Soc. Transl. Ser. 2},
VOLUME = {196},
PAGES = {183--198},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1999},
MRCLASS = {14N35 (14E30 53D45)},
MRNUMBER = {2001h:14074},
MRREVIEWER = {Holger Spielberg},
ZBLNUMBER = {0952.53001},
} -
[yR1] Y. Ruan, "Quantum cohomology and its application," in Proc. Internat. Congress of Mathematicians, Vol. II, 1998, pp. 411-420.
@inproceedings {yR1, MRKEY = {1648091},
AUTHOR = {Ruan, Yongbin},
TITLE = {Quantum cohomology and its application},
BOOKTITLE = {Proc. {I}nternat. {C}ongress of {M}athematicians, {V}ol. {\rm II}},
VENUE={{B}erlin, 1998},
JOURNAL = {Doc. Math.},
FJOURNAL = {Documenta Mathematica},
YEAR = {1998},
PAGES = {411--420},
ISSN = {1431-0635},
MRCLASS = {53D45 (14N35)},
MRNUMBER = {2000b:53119},
MRREVIEWER = {V{é}ronique Lizan},
ZBLNUMBER = {0972.53054},
} -
[yR2] Y. Ruan, "The cohomology ring of crepant resolutions of orbifolds," in Gromov-Witten Theory of Spin Curves and Orbifolds, Providence, RI: Amer. Math. Soc., 2006, vol. 403.
@incollection{yR2,
author = {Ruan, Yongbin},
TITLE = {The cohomology ring of crepant resolutions of orbifolds},
BOOKTITLE={Gromov-Witten Theory of Spin Curves and Orbifolds},
SERIES={Contemp. Math.},
VOLUME={403},
PUBLISHER={Amer. Math. Soc.},
ADDRESS={Providence, RI},
YEAR={2006},
MRNUMBER={2234886},
} -
[Wang1] C. Wang, "On the topology of birational minimal models," J. Differential Geom., vol. 50, iss. 1, pp. 129-146, 1998.
@article {Wang1, MRKEY = {1678489},
AUTHOR = {Wang, Chin-Lung},
TITLE = {On the topology of birational minimal models},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {50},
YEAR = {1998},
NUMBER = {1},
PAGES = {129--146},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {14E05 (14E30 14F42 14G99)},
MRNUMBER = {2000m:14012},
MRREVIEWER = {Alessio Corti},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214510049},
ZBLNUMBER = {0956.14015},
} -
[Wang2] C. Wang, "$K$-equivalence in birational geometry," in International Congress of Chinese Mathematicians, Sommerville, MA: Internat. Press, 2004, vol. 4.
@incollection{Wang2,
author = {Wang, Chin-Lung},
TITLE = {$K$-equivalence in birational geometry},
BOOKTITLE={International Congress of Chinese Mathematicians},
VENUE={Grand Hotel, Taipei 2001},
SERIES={New Stud. Adv. Math.},
VOLUME={4},
PUBLISHER={Internat. Press},
ADDRESS={Sommerville, MA},
YEAR={2004},
ARXIV={math.AG/0204160},
} -
[Wang3] C. Wang, "$K$-equivalence in birational geometry and characterizations of complex elliptic genera," J. Algebraic Geom., vol. 12, iss. 2, pp. 285-306, 2003.
@article {Wang3, MRKEY = {1949645},
AUTHOR = {Wang, Chin-Lung},
TITLE = {{$K$}-equivalence in birational geometry and characterizations of complex elliptic genera},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {12},
YEAR = {2003},
NUMBER = {2},
PAGES = {285--306},
ISSN = {1056-3911},
MRCLASS = {14E05 (58J26)},
MRNUMBER = {2003j:14015},
MRREVIEWER = {Lev A. Borisov},
ZBLNUMBER = {1080.14510},
} -
[Witten] E. Witten, "Phases of $N=2$ theories in two dimensions," Nuclear Phys. B, vol. 403, iss. 1-2, pp. 159-222, 1993.
@article {Witten, MRKEY = {1232617},
AUTHOR = {Witten, Edward},
TITLE = {Phases of {$N=2$} theories in two dimensions},
JOURNAL = {Nuclear Phys. {\rm B}},
FJOURNAL = {Nuclear Physics. B},
VOLUME = {403},
YEAR = {1993},
NUMBER = {1-2},
PAGES = {159--222},
ISSN = {0550-3213},
CODEN = {NUPBBO},
MRCLASS = {81T40 (58Z05 81T60)},
MRNUMBER = {95a:81261},
MRREVIEWER = {Tsou Sheung Tsun},
DOI = {10.1016/0550-3213(93)90033-L},
ZBLNUMBER = {0910.14020},
}
Full Article